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Learning algorithms for manipulator control Huscroft, Charles Kevin 1984

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LEARNING  ALGORITHMS FOR  MANIPULATOR CONTROL By CHARLES K E V I N HUSCROFT B . A . S c . ( H o n s ) , The U n i v e r s i t y  of B r i t i s h  Columbia, 1979  A THESIS SUBMITTED I N PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF A P P L I E D SCIENCE in THE FACULTY OF GRADUATE (Department of E l e c t r i c a l  We a c c e p t to  this  t h e s i s as  the required  STUDIES Engineering)  conforming  standard  THE UNIVERSITY OF B R I T I S H COLUMBIA October © Charles  1984  K e v i n H u s c r o f t , 1984  In p r e s e n t i n g  this thesis  r e q u i r e m e n t s f o r an of  British  it  freely available  in partial  advanced degree a t  Columbia, I agree that for reference  agree t h a t permission  understood for  that  h i s or  be  her  s h a l l not  1Y3  Date  and  study.  I  October 17,  1984  the  of  further this  Columbia  thesis  head o f  this  my  It is thesis  a l l o w e d w i t h o u t my  E l e c t r i c a l Engineering  The U n i v e r s i t y o f B r i t i s h 1956 Main Mall V a n c o u v e r , Canada  V6T  s h a l l make  representatives.  permission.  Department Of  University  Library  g r a n t e d by  be  the  the  copying or p u b l i c a t i o n  f i n a n c i a l gain  the  for extensive copying of  f o r s c h o l a r l y p u r p o s e s may d e p a r t m e n t o r by  f u l f i l m e n t of  written  Abstract A  method of r o b o t  algorithms of  are  u s e d t o l e a r n sum  manipulator  learned using  dynamics  and  c o n t r o l i s proposed  of p o l y n o m i a l s  kinematics  r e l a t i o n s h i p s are u t i l i z e d the  learning  technique is  manipulator; Rates compared variate  manipulator  of R e s o l v e d  achieved  without  functions.  Interference  the  Acceleration recourse  of c o n v e r g e n c e o f s e v e r a l when l e a r n i n g e s t i m a t e s  representations  relationships.  to c o n t r o l  h e n c e t h e name S e l f - L e a r n e d  whereby  to  manipulator  Control. analysis  of  the  algorithms  are  of v a r i o u s n o n - l i n e a r ,  multi-  M i n i m i z a t i o n i s found  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  K l e t t C e r e b e l l a r Model.  Simplification  Minimization i s described.  Interference  Minimization,  Such  Control.  learning  Interference  The  Identification  of the A  to  be  and  the  implementation  variant,  i s introduced that  of  Pointwise  i s suitable  for  certain applications. Self-Learned coordinates  is  manipulator. of t h e  I t i s shown t h a t sum  be  to that achieved learned without  Further adaptation  research t o t o o l mass,  e s t i m a t o r s and  for  specification a  of p o l y n o m i a l s  adequate  to  outlined  implementation  enhancement of the K l e t t  to  Cartesian two  link  representations direct  position  achieve  using their a n a l y t i c a l  a n a l y s i s of t h e is  in  simulated  i n v e r s e k i n e m a t i c s and  r e l a t i o n s h i p s are  comparable can  demonstrated  inverse dynamics,  kinematics  and  C o n t r o l with path  control  counterparts  manipulator. achieve o f sum  of  automatic polynomials  C e r e b e l l a r Model.  Table  of  Contents  Abstract  i i  List  of F i g u r e s  v i i  List  of Tables  x i i  Acknowledgements  xiv  1.  Introduction  2.  Interference Minimization Algorithms 2.1  2.2  2.3  2.4  and Related  6  Learned F u n c t i o n a l Estimation of P o l y n o m i a l s  using  Choice of Test of A l g o r i t h m s  Sums 6  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 Interference Minimization Relationship to Similar Algorithms  Learning  7  Learning 9  Conditions  f o r Comparison 13  2.5  Comparison  2.6  Implementation Considerations for Interference Minimization  28  A p p l i c a t i o n s where P o i n t w i s e Minimization i s Appropriate  32  2.7  2.8 3.  1  of the Learning  33 o f a Two  Link  Resolved Acceleration Control C a r t e s i a n Path S p e c i f i c a t o n  3.2  A Two  Link  3.3  Simulation  3.4  Path  18  Interference  Summary  Self-Learned Control 3.1  Algorithms  Manipulator  .. 34  with 34  Manipulator  41  of the Manipulator  47  Specification  53  iii  3.5  Manipulator Control using 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 Inverse Kinematics 3.5.1  Ideal  Open  3.5.2  Ideal  Closed  3.5.3  R e a l i z a b l e Closed Loop C o n t r o l Standard A n a l y t i c a l Control  3.5.4  3.6  Control  Loop  61  Control  65 67  Choice of Parameters D e f i n i n g Standard A n a l y t i c a l Control  72  A d e q u a c y 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 of the I n v e r s e Dynamics Inverse Kinematics 3.6.1  and 87  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 of the Inverse Dynamics  87  3.6.2  Pre-Learning  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 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 the C a r t e s i a n Inverse Dynamics  93  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 Representation of the C a r t e s i a n Inverse Dynamics 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  95  Pre-Learning Dynamics  98  3.6.4  3.6.5  3.6.6  3.7  Loop  61  of  of  the  Inverse  Dynamics  the Cartesian  Closed Loop C o n t r o l u s i n g Learned Cartesian Inverse  ..  Inverse  the PreDynamics  ....  A d e q u a c y o f a Sum o f P o l y n o m i a l s Representation of the D i r e c t Position Kinematics 3.7.1  3.7.2  3.7.3  100  102  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 Representation of the D i r e c t Position Kinematics  103  Pre-Learning Kinematics  107  of  the Direct  Position  C l o s e d Loop C o n t r o l u s i n g the Pre-Learned C a r t e s i a n Inverse Dynamics and P r e - L e a r n e d D i r e c t Position Kinematics  iv  112  3.8  Self-Learning Dynamics 3.8.1  3.8.2  3.9  3.9.2  Inverse  A Method f o r S e l f - L e a r n i n g C a r t e s i a n Inverse Dynamics Closed Loop C o n t r o l using Learned Cartesian Inverse of  the D i r e c t  of  the 120  the S e l f Dynamics ....  139 of  the 139  Closed Loop C o n t r o l using the S e l f Learned C a r t e s i a n Inverse Dynamics and Self-Learned Direct Position Kinematics  Summary  5.  Suggestions  of  this  Thesis  for Future  171  Research  Investigation  of  174 Self-Learned  Control  5.2  5.3  155 165  Contributions  Further  133  Position  A Method f o r S e l f - L e a r n i n g Direct Position Kinematics  4.  5.1  the Cartesian  120  Self-Learning Kinematics 3.9.1  3.10  of  174  5.1.1  Control  o f a More  5.1.2  Adaptation  to Tool  Implementation Estimators  of  Sum  Digital  5.2.1  Stochastic  Learned  5.3.3  Input  174 175  Polynomials  Implementation  Computer  Proposed M o d i f i c a t i o n s C e r e b e l l a r Model  5.3.2  .  Mass  of  Computer  The  Manipulator  178  5.2.1  5.3.1  Complex  Klett  Implementation  to the  Cerebellar  186  Klett  Model  Splitting  v  ....  190  Orthogonalization  Variable  179  190 193 199  Bibliography A p p e n d i x A: A p p e n d i x B:  206 A n a l y t i c a l Kinematics Manipulator  o f t h e Two  A n a l y t i c a l D y n a m i c s o f t h e Two .Manipulator  Link 211  Link 215  A p p e n d i x C:  C l o s e d Loop 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 Inverse Dynamics a f t e r 200, 400, 600, 800 and 1000 T r a i n i n g Paths 218  A p p e n d i x D:  View 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 200, 400, 600, 800, 1000, 1400, 1800, 2200, 2600, 3000, 3400, 3800, 4200, and 4600 T r a i n i n g Paths  vi  224  List 2.1  2.2  of  Figures  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 chosen 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 function 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 a n d v=3  15  C o n v e r g e n c e r a t e s f o r M e t h o d 5, t h e C e r e b e l l a r M o d e l , a s a f u n c t i o n o f u f o r t h e c a s e s s=1 a n d v=3, s=3 a n d v=3, a n d s=3 a n d  17  2. 3  Convergence  rates  for  various  methods  when  s= 1 . .  20  2. 4  Convergence  rates  for  various  methods  when  s= 2 . .  21  2. 5  Convergence  rates  for  various  methods  when  s= 3 . .  22  2. 6  Convergence  rates  for  various  methods  when  s= 4  ..  23  2. 7  Convergence  rates  for  various  methods  when  s= 5  ..  24  2. 8  Convergence  rates  for  various  methods  when  s= 6  ..  25  2. 9  e r r o r , Af, as a Reduction of estimation 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 o i n t s f o r t h e c a s e s=3 a n d 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 iteration interval)  26  P a t t e r n of zero and non-zero elements of matrix P f o r t h e c a s e s=3 a n d v=3 (+ = p o s i t i v e , - = n e g a t i v e , 0 = zero)  29  T o t a l number o f e l e m e n t s , number elements and order of matrix P w h e r e s=3  31  2.10  2.11  of non-zero f o r cases  3.1  A two l i n k  3.2  Block  3.3  Standard  path  line  1 in Cartesian  3.4  Standard  path  line  1 in joint  3.5  Standard  path  circle  1 in Cartesian  3.6  Standard  path  circle  1 in joint  3.7  View o f standard path ( t i c k s mark i n t e r v a l s  manipulator  diagram  42  of manipulator  simulation  coordinates  coordinates coordinates  coordinates  circle 1 o f 0.2 s e c )  vii  program  . 48  . . . 55 56 . 58 59  60  3.8  Ideal  3.9  Torque  3.10  Error correcting control  3.11  open  profile  Standard  3.12 S t a n d a r d that 3.13  t c  a  i  action  1  64  of i d e a l  closed  1  analytical control  of l i n e  1,  c  '  0  2  s  e  of line  t c  a  i  analytical control = 0 c  -  0  4  s  e  1,  except 7  of l i n 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 t h a t USEOBS=.TRUE  of l i n 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 t h a t k =64 a n d k = l 0 2 4  of l i n e  3.19  Standard a n a l y t i c a l control t h a t k 4 and k =4  of l i n e  Standard a n a l y t i c a l control that k 8 and k =l28  of l i n e  Standard a n a l y t i c a l control t h a t ^ = 3 2 and k =32  of l i n e  1,  1 =  5  except 77  1,  except 78  1,  except 80  1,  except 82  2  1 =  3.21  except  c  Standard a n a l y t i c a l control t h a t PRDICT=.FALSE  1  3  74  of l i n e  3.20  1,  3  c  Standard a n a l y t i c a l control t h a t EXACT=.TRUE  3.16  except 7  Standard that  71'  c  of l i n e  0  loop 66  of l i n e  = 0  a  62  1  Standard a n a l y t i c a l control that i ' sec c  3.15  for line  of l i n e  analytical control  = 0  t  3.14  loop control  1,  except 83  2  1,  except 84  2  1,  except 85  2  3.22 R e d u c t i o n o f e s t i m a t i o n error during pre-learning of t h e i n v e r s e dynamics ( e s t i m a t i o n error 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  Reduction of estimation error during pre-learning of t h e C a r t e s i a n inverse dynamics (estimation 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 )  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 learned Cartesian inverse dynamics 3.25  View o f c i r c l e 1 u s i n g position kinematics  the derived  99  the pre101 direct 106  vi  ii  3.26  3.27  3.28  3.29  3.30  3.31  3.32  3.33  3.34  3.35  3.36  3.37  3.38  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 (estimation 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  View of c i r c l e 1 u s i n g position kinematics  110  the pre-learned  direct  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 direct p o s i t i o n kinematics 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 using the pre-learned Cartesian inverse dynamics and p r e - l e a r n e d d i r e c t position kinematics  114  View of 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 of c i r c l e 1 using the pre-learned Cartesian inverse dynamics and p r e - l e a r n e d d i r e c t position kinematics  115  Closed loop pre-learned pre-learned  118  c o n t r o l of l i n e 1 using the C a r t e s i a n i n v e r s e dynamics and direct position kinematics  Closed loop c o n t r o l of l i n 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 pre-learned d i r e c t position kinematics with ADJVIS=.TRUE.  119  C h o i c e s of a_ and v f o r f i r s t 250 t r a i n i n g paths 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 inverse dynamics  127  V i e w 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 of the C a r t e s i a n i n v e r s e dynamics  128  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 p l a c e 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 inverse dynamics  129  Path e r r o r s during s e l f - l e a r n i n g C a r t e s i a n inverse dynamics  130  m  of the  Maximum o u t o f b o u n d s e x c u r s i o n s d u r i n g selfl 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  131  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 inverse dynamics as a f u n c t i o n of the number o f t r a i n i n g p a t h s u s e d  134  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  135  ix  final  3.39  3.40  3.41  3.42  3.43  3.44  3.45  3.46  3.47  3.48  3.49  3.50  Closed loop c o n t r o l of c i r c l 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  final 136  Average torque estimation 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 o f t h e n u m b e r o f t r a i n i n g p a t h s u s e d ...  138  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 p l a c e during s e l f - l e a r n i n g of the direct p o s i t i o n kinematics  144  Path estimation e r r o r s during s e l f - l e a r n i n g the d i r e c t p o s i t i o n kinematics  145  View of c i r c l e 1 using the f i n a l d i r e c t p o s i t i o n kinematics  of  self-learned 146  View of c i r c l e 1 u s i n g the smoothed, final self-learned direct position kinematics  154  View of c i r c l e 1 using the f i n a l self-learned direct p o s i t i o n kinematics during closed loop c o n t r o l of c i r c l e 1 using the f i n a l self-learned C a r t e s i a n i n v e r s e dynamics and f i n a l self-learned direct position kinematics  156  View o f c i r c l e 1 during c l o s e d loop c o n t r o l of circle 1 using the f i n a l self-learned Cartesian i n v e r s e dynamics and f i n a l s e l f - l e a r n e d direct position kinematics  157  View o f c i r c l e 1 u s i n g the smoothed, f i n a l selflearned direct position kinematics during closed loop c o n t r o l of c i r c l e 1 using the f i n a l selfl 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 self-learned direct position kinematics ...  160  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 circle 1 using the final self-learned Cartesian i n v e r s e dynamics and smoothed, f i n a l selflearned direct position kinematics  161  Closed loop c o n t r o l of line 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 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 position kinematics  163  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 position k i n e m a t i c s w i t h ADJVIS=.TRUE  164  x  5.1  5.2  B l o c k d i a g r a m of d i g i t a l computer o f a sum o f p o l y n o m i a l s e s t i m a t o r  implementation 184  S t o c h a s t i c c o m p u t e r c i r c u i t f o r summing t h e p r o d u c t s o f two p a i r s o f i n p u t v a r i a b l e s  189  5.3  Block  diagram  of  cerebellar  190  5.4  Block  diagram  of  the  5.5  Schematic of Granule c e l l - G o l g i c e l l network a) s h o w i n g 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 could occur b) s h o w i n g l u m p e d a m p l i f i c a t i o n s  194  5.6  Example  202  5.7  Representations  5.8  E n c o d i n g of splitting  5.9  of  input  Klett  variable of  joint  system Cerebellar  Model  splitting  sin(27rz) angle  191  by  • ••• input  202  variable 205  Example of p r o d u c t s of s p l i t representing joint angle  xi  input  variables 205  List 2.1  Summary  of  of  Tables  learning algorithms  13  * 2.2  Step size factors, C e r e b e l l a r Model  u,  f o r Method  Data  d e f i n i n g standard  linear  3.2  Data  d e f i n i n g standard  circular  3.3  Path  error  using  ideal  open  3.4  Path  error  using  ideal  closed  3.5  Path  error  using  standard  3.6  Path e r r o r using except f o r noted  3.8  3.9  3.10  3.11  3.12  3.14  3.15  paths  loop  .-  54  paths  57  control  loop  63  control  analytical  standard a n a l y t i c a l variations  control  67 ....  70  control, 86  C o e f f i c i e n t s f o r the 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 inverse dynamics f u n c t i o n f o r torque  92  C o e f f i c i e n t s f o r the 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 inverse dynamics f u n c t i o n f o r torque  93  Path e r r o r and torque e r r 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 f o r c l o s e d control  102  Path estimation error position kinematics  using  pre-learned loop  the derived  direct 105  Path 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  C o e f f i c i e n t s f o r the 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 position kinematics functions for positions x  3.13  the 18  3.1  3.7  5,  1  and  *2  Path estimation direct position  1  error using kinematics  the  1  1  pre-learned 112  Path e r r o r using the pre-learned 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 position kinematics for closed loop control  113  Path e r r o r and torque 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 C a r t e s i a n i n v e r s e dynamics f o r closed loop control  139  xi i  3.16  Path estimation error using the f i n a l self-learned direct position kinematics  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 a n d 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 representations 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 functions for p o s i t i o n s x and 1  3.18  3.19  Path e s t i m a t i o n e r r o r u s i n g t h e smoothed, self-learned direct position kinematics  final 152  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, final 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 representations 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 functions for p o s i t i o n s x and x 1  2  3.20 P a t h e r r o r u s i n g t h e f i n a l self-learned C a r t e s i a n i n v e r s e dynamics and f i n a l self-learned direct position kinematics in c l o s e d loop c o n t r o l 3.21  3.22  5.1  Path e r r o r using the f i n a l self-learned C a r t e s i a n i n v e r s e dynamics and smoothed, self-learned direct position kinematics in closed loop c o n t r o l Summary schemes Number s,  of path  errors  using  various  148  153  158  final 159  control 170  of terms,  and various  m,  f o r systems  numbers  of input  of various  orders,  variables, v  . . . . 181  5.2  M u l t i p l i c a t i o n s required f o r estimates of various orders of the C a r t e s i a n inverse dynamics f o r manipulators 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  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 estimates of v a r i o u s orders of the d i r e c t p o s i t i o n kinematics f o r manipulators 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  xi ii  Acknowledgements I for  his  would many  would  like  also ideas  like he  to  wife,  constant  and  I  dedicate  I  will  my  of  supervisor,  fellow  always research  frequent  J.  Lawrence,  support.  Clark,  I  for  the  source  of  conversations.  has  daughter,  Alicia  Estrella,  nephew  Lawrence  t h e s i s to  been  a  constant  has  been  a  my  Wyatt  Telling,  remember. has  been  Research  of  James  D.  and  Estrella,  Scholarship  University  Peter  assistance  student,  our  Dr.  inspiration.  this  Engineering  Postgraduate  thank  Dulce  source  This  the  my  offered during  encouragement  and  thank  e n t h u s i a s t i c encouragement,  My  whom  to  to  British  supported  by  the  Council  of  Canada  and  Grant  i t s author Columbia  Assistanceship.  xiv  i n the  Natural  form  Sciences  through #A4924, of  a  and  a by  Teaching  1  There field  of  robotics.  analysis The  is currently  of  goal  of  algorithms linear,  analysis The proposed  robot this  the  by  [26].  earlier  work  [42,49].  by on  The  of  motor  manipulator  control. the  a  family  of  work  previous and  was  of  i t is  to  proposed  Klett  [31]  such  learning  body  by  draws  in maintaining  as upon  machines  posture  [1,26,38];  reasonable  C e r e b e l l a r Model  intended  Model  and  is  and  Cerebellar  Marr  the  non-  recourse  models  of i t  the  cerebellar  is involved  learning  manipulator.  the  other  avoid  dynamics  without the  the  manipulators.  learn  machine  and  activities  Klett  robot  done  learning  Thus  of  to  in  d e s i r e to  Furthermore, be  perceptrons,  cerebellum  of  can  [1,2]  the  control  d e s c r i b i n g the  this  The  to  ability  kinematics  for  Albus  coordinating  application  or  extension  proposed  their  learning  point  an  examine  functions  dynamics  Klett  i s due  manipulator.  starting  represents those  a  in adaptive  kinematics  i s to  demonstrate  that of  this  and  thesis  multi-variate  show  interest  Largely,  this  of  much  dynamics  and  kinematics to  the  INTRODUCTION  to  in learned  and i.e.  consider manipulator  control. Currently, individual oriented  joint  into  positions,  control  typically  industrial  servo  coordinate  transformed joint  most  a  control.  system path time  robot  such  manipulators  Path as  specifications  neglects  task.  coupling  1  based  in a  on  task  C a r t e s i a n c o o r d i n a t e s must  specifications consuming  are  i n terms  of  Individual  dynamic  effects  be  successive joint  servo  and  thus  performance  i s limited.  Manipulator use  of  analytical  i s Resolved  Acceleration  [ 3 0 ] . The  require  disadvantage  analysis  manipulator  of  the  the  dynamics  such  as l i n k  moments  of  may  the manufacturer Adaptive  of manipulator  parameters  [ 2 9 ] . One  of  the  recent  observations Performance  require  a priori  Intuitively, inefficient learning,  the  i t  model  place  by  i s  et  each  i s often a d i f f i c u l t  task.  kinematics  i n the formulation centers  to obtain  of  the  of gravity by a n y o n e  i s  and other  for Linear  i s continually  information  and the  the  i n f o r m a t i o n from  2  Linear  valid  model  based  on  manipulator does  not  o r measurement  of  many  mathematical  on-line  operations.  i t i s continually  old  Adaptive  method  Controller  forgetting  the  manipulator  and r e s u l t i n g  as  as the manipulator  from  of  on-line,  Unfortunately,  although  incorporating  estimated  good  avoid  locally  of the manipulator  Adaptive  i s changed  i s the  [27]. A  torques  quite  to  and measurement  a n d Guo  dynamics  i s  proposed  technique  necessary  because  valid  forgetting  such  analysis  are  by L u h  an  of  been  analysis  parameters.  operations  manipulator;  they  masses,  of a p p l i e d  motion.  manipulator  used  have  by K o i v o  manipulator  This  make  that  and  be d i f f i c u l t  techniques  proposed  of a  that  of the manipulator.  necessity  Controller  proposed  techniques  dynamics  lengths,  been  C o n t r o l as proposed  of such  parameters  inertia  have  dynamics  t o be s o c o n t r o l l e d .  Furthermore,  than  techniques  the complete  example al  control  would  seem  adapting,  well.  The  moves from  new  to  i . e .  locally  place  observations  observations.  Thus  be  to and upon  repeating benefit  a  path,  the  Adaptive  from the p r e v i o u s  Self-Learned  Linear  Controller  does  not  experience.  Control  as p r o p o s e d i n t h i s  thesis  differs  f r o m 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 m o d e l of the  manipulator  respect  it  is  Articulation of  similar  to  is  learned.  the  of  than  our  Due  Model use  functional  fewer  t h e method p r o p o s e d by A l b u s  this  to the  learned  method r e q u i r e s f a r  input v a r i a b l e s .  In  Cerebellar  f u n c t i o n s as the b a s i s of our  however,  coefficients encoding  somewhat  kinematics  C o n t r o l l e r p r o p o s e d by A l b u s [ l , 2 ] .  continuous  estimates,  d y n a m i c s and  and  weighting no  Once t h e d y n a m i c s and  special kinematics  h a v e been l e a r n e d t o a d e g r e 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 , further  l e a r n i n g need take p l a c e .  the manipulator  d y n a m i c s and  T h i s assumes, of c o u r s e ,  kinematics  a r e not  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 p r i n c i p l e was  no that  changing.  learning algorithm  used  in this thesis,  d e r i v e d by K l e t t a s  step  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 M o d e l  an  intermediate  [26].  Chapter  examines t h e c o n v e r g e n c e 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 related  l e a r n i n g a l g o r i t h m s s u c h as t h e G r a d i e n t  and  Learning  the  number  Minimization counterparts  Identification  [12,39,50],  Method  A method  of c a l c u l a t i o n s r e q u i r e d t o implement is  introduced.  Also  of s e v e r a l of t h e s e  introduced  of  2  and  [36,57] reducing  Interference are  Pointwise  a l g o r i t h m s t h a t have u t i l i t y  in  certain applications. In  chapter  applied to achieve manipulator.  3 the aforementioned  learning  algorithms  S e l f - L e a r n e d c o n t r o l o f a s i m u l a t e d two  S e c t i o n 3.1  discusses Resolved  3  are link  Acceleration Control  using 3.2 of the  path  specification  p r o v i d e s t h e a n a l y t i c a l d y n a m i c s and t h e two two  path  l i n k manipulator.  l i n k manipulator  specifications  analytical  are  implementation Standard  Analytical  benchmark  dynamics  and  manipulator a  Pre-Learn  be  is  Cartesian  without  kinematics.  to  the  information. Learned vision  The  a  realistic  Control that a  we  performance  with Self-Learned  kinematics  Furthermore,  a  of  the  control.  the  link  polynomials  portion  of  the  i t i s shown t o be p o s s i b l e t o  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 inverse  inverse  two  a s sums o f  sizable  call  d y n a m i c s and  analytical  using  analytical  i n s e c t i o n s 3.8 can  be  dynamics  the  direct and  3.9  Self-Learned or  analytical  In c o n t r o l using 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  manipulator Section  direct system  of  This establishes  of  test  link  i t i s shown t h a t t h e C a r t e s i a n  d y n a m i c s a s shown i n s e c t i o n observe  standard  two  simulation  position  how  k i n e m a t i c s of the  that these representations  recourse  outline  the  k i n e m a t i c s as a g u i d e . F i n a l l y , shown  kinematics  3.5  Acceleration  consisting  sum  how  In  represented adequately  reach.  these  analytical  3.7  direct  can  manipulator's  it  a  Control.  and  workspace  position  in  3.4  section  section  f o r subsequent comparison  s e c t i o n s 3.6  over  generated.  of R e s o l v e d  and  s i m u l a t e d and  analytical  used  analytical  S e c t i o n s 3.3  was  were  d y n a m i c s and  manipulator  In  i n C a r t e s i a n c o o r d i n a t e s and  and  3.9  position  3.8,  a vision  system  i s assumed t o  provide error correcting  shows t h a t u t i l i z a t i o n  of  feedback the  k i n e m a t i c s permits replacement  once l e a r n i n g  is  Selfof  the  complete.  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  4  chapter  4 and c h a p t e r future to  a  research. real  extend  to  be a d j u s t e d  the  there having  Self-Learning  of a t o o l  investigation  required  sum  modifications  of  detailed  more  than  2 degrees  t o compensate o f a n y mass  of polynomials  to apply  the Cartesian  i s warranted  the Klett  suggestions  i s the need  to allow  automatically  manipulator  Secondly,  several  First,  manipulator  to  the  5 offers  plausibility.  5  and  dynamics  given  Model  Finally, may  to  range.  the implementation  estimators.  Cerebellar  of freedom  inverse  a  of  Self-Learning  f o r the attachment  within  into  f o r areas  of  proposed  increase  i t s  2  2.1  INTERFERENCE MINIMIZATION AND  RELATED LEARNING ALGORITHMS  LEARNED  USING  In  FUNCTIONAL  adaptive  estimate  In  estimation  is  is  Widrow the  linear was  [57]  method  of  Learning  authors  time  These  non-linear  systems  of  a  to  [26,10,11]. functions  methods [26],  and  of  and  Odile  [14],  have  been  to  signal functions  variables  in for  estimate  of  the  a  other and  Billings  function  of  can  be  arbitrary  learning  [50].  by  and  They  also  variables  of  application  for  Bitmead  [25]  [39,50,57],  used  technique  Sherman  investigated  Johnson  used  Method  Identification  and  [10,11],  interested  several  been  Eweda  continuous  are  have  Roy  functional  formulated  Learning  by  learning  Another was  to  polynomials  Gradient  Babu  estimate  We  [39],  a  of  I d e n t i f i c a t i o n that  to  Klett  sum  the  learn  whereby  filtering.  Noda  Chan  generalized  adaptive  a  is  Nagumo a n d  techniques  samples  on  to  through  techniques  by  as  [8],  study  POLYNOMIALS  necessary  relationships  technique  for  related  such  Anderson [7]..  such  SUMS OF  is often  based  al  and  we  learned  et  extended  These  thesis  One  systems  it  functional  this  representation. by  control,  unknown  technique.  ESTIMATION  non-linear  in  manipulator  control. This which They rates  thesis  the  previously  thus  form  of  the  simplifying algorithm  describes  are  a  improved  mentioned  family various  the  an  of  learning  algorithms  learning  algorithms  implementation  discussed.  6  are  algorithms. are of  algorithm  simplifications. The  compared the  of  convergence and  improved  factors learning  2.2  AN  IMPROVED L E A R N I N G  This  thesis  estimate  a  formed f  i s concerned  f u n c t i o n as a  Kolmogorov-Gabor is  / c  p  sum  -  INTERFERENCE  with  algorithms  of polynomials,  [ 18,50] p o l y n o m i a l s .  as a weighted  = I w  ALGORITHM  sum  The  MINIMIZATION that  specifically  estimate  of polynomial  learn  the  of a f u n c t i o n  terms,  (z) = w p  (2.1)  T  / c  to  k  The  basis  polynomial  {p.(z)J  =  terms  n z. i i=0  {  }  e  *  are the s e t ,  (2.2)  where, v Z  e.  i=0  z For  =  =  Q  e. a n  integer  (2.3)  1  (p .(z)} A  space  (2.4) i f s=2 =  a n d v=2  { 1, z  spanned  variables, such  ,  1  example,  The  s  1  and  by  ]  t  z , 2  this  then  the polynomial  z ^  z  2 2  , z ^  set i s given  s, t h e system  a polynomial  f  i s given  order.  2  }  by v ,  The  terms a r e , (2.5)  t h e number  number  of  of terms,  input m,  in  by,  (s+v)! m =  (2.6) s!v!  An  estimate  linear the  formed  functions.  weight  estimate  in this Learning  vector  at a  corresponds  to  manner takes  series  can mimic place  by  iteratively  of t r a i n i n g  the target  multivariate,  points  function  non-  adjusting until  the  throughout  the  space. An At  each  improved training  learning point  algorithm  the weights  7  c a n be d e r i v e d  are adjusted  as  follows:  t o e l i m i n a t e the  error  in  estimate  the  estimate  at other  Interference minimizing  the  change  i n the  call  the  points  Minimization. the T  This  2  5S  the  and  Af  i s given  s t r a t e g y can  be  enforced  Aw  e r r o r i n the  technique,  where S i s t h e  domain  i s the change i n the w e i g h t  e s t i m a t i o n . The  e r r o r i n the  vector estimate  by,  where f i s t h e c  by  (2.7)  Af = f - f *  of  the  algorithm,  T  input v a r i a b l e s , i s the  We  + a(Aw p - Af)  the Lagrange m u l t i p l i e r  of  space.  in  function,  c = / (Aw p) S using  while minimizing  with  (2.8) target  respect  weight adjustment  function.  t o Aw  and  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  a t o z e r o and  solving  yields  the  formula,  AfP~ p 1  Aw  =  (2.9)  P P where, P = / pp S  The the  matrix  P  6S  T  (2.10)  P is real,  inverse, P  1  can work  be  polynomials other  Minimization  step  generalized we  consider form  previously  polynomials  and  positive definite  and  thus  exists.  Interference intermediate  s y m m e t r i c and  i n the  was  [26] as  an  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  t o a l l o w use in the  depth  of o t h e r the  basis set.  described permits  d e r i v e d by K l e t t  basis  specific  sets.  learning algorithms  8  of t h e  this  where  K-G  comparison  with  case  This e l i c i t s  simplification  In  based  on  implementation  K-G of  Interference Minimization  2.3  RELATIONSHIP TO Several  can  Interference matrix  P  does  adjustment matrix  P  change  Aw,  the  effect  u s i n g K-G  order  simplifications  of  i t can  be  magnitude  seen of  point  is  that  the  This  sub-optimal  weight  and  If  the the  eliminated,  points  in  the  learning algorithm  formulated  polynomials,  the  formula,  still  of the a d j u s t m e n t a t o t h e r  minimized.  first  the  weight adjustment  I d e n t i f i c a t i o n , w h i c h was  using  in  i t only changes the d i r e c t i o n .  a training  at  be  (2.9)  error  space i s not  [39]  In  from the  the  2.6.  previously described to  i s deleted  estimation  Learning  considered  not  vector 1  however,  be  algorithms  Minimization.  1  i s shown i n s e c t i o n  S I M I L A R LEARNING ALGORITHMS  learning  literature  as  by Nagumo and  by Roy  and  is Noda  Sherman  [50]  polynomials, Mp  Aw  = -=P P  In  (2.11) the denominator s e r v e s  adjusts  the  step s i z e .  the estimate the  When c h a n g i n g t h e w e i g h t s ,  i s exactly eliminated regardless  then the d i r e c t i o n of the  changed;  eliminates  the  one  Aw  still  the  of the  i s replaced  error in  location  by a  e r r o r at the  weight  training  of  positive  set  is  that  point.  The  becomes,  = uAfp  algorithm  that  weight adjustment vector  moves t o w a r d s a new  estimation  adjustment formula  This  as a s c a l i n g v a r i a b l e  t r a i n i n g p o i n t . I f the denominator  constant, not  (2.11)  (2.12) i s known a s t h e G r a d i e n t  9  Method  and  has  been  discussed Odile  by  authors  [ 1 4 ] . The  adjustments training  gain  are  not  points  choosing  u such  s u c h a s Widrow e t a l [ 5 7 ] a n d Eweda factor  u  must  be  chosen  so l a r g e a s t o cause  within  the space.  This  such  divergence can  be  just  (2.13)  i s even  are  less optimal  than  r e q u i r e d a t most t r a i n i n g  (2.11) points  since  many  the  space  in  t o eliminate the estimation e r r o r at the t r a i n i n g variants  matrix.  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  b a s i s f u n c t i o n s such that P  selects  by  that,  algorithm  Two  any  ensured  T  iterations  that  at  u < 2 min{l/(p p)} S This  and  i s equal  1  follow t o the  point. i f one identity  One method i s t o f o r m t h e e s t i m a t e a s ,  f = w Qp = w q T  (2.14)  T  where, Q = [/ p p S The m a t r i x  T  6S]"  1 / 2  Q i s real,  of t h e v e c t o r  = P(p)"  symmetric  a n d p o s i t i v e d e f i n i t e . The t e r m s  q are a set of orthonormal polynomials.  a basis vector, the matrix P(q)  (2.15)  l / 2  = / qq S  T  such  P becomes,  6S = / Q p p Q S T  The w e i g h t a d j u s t m e n t  With  formula  T  SS = P "  l / 2  PP  _ 1 / 2  = I  (2.16)  now becomes,  Mq  Aw = -=q q This  (2.17)  method i s e q u i v a l e n t  previously  in  (2.9);  estimation  error  to Interference Minimization  the  weight  at the training  10  adjustment point  while  as given  eliminates  the  minimizing  the  change only  i n the e s t i m a t e at other p o i n t s i n the space. represents  formula  at  simplification  orthonormal  Orthogonal  of  t h e e x p e n s e o f a more  involving  As  a  We  formula  this  method,  i n (2.17) s e r v e s as a and  can  be  scaling  replaced with a  constant; =  (2.18)  uAf q  To a v o i d d i v e r g e n c e must c h o o s e u s u c h u  adjustment  estimation call  really  Interference Minimization.  v a r i a b l e t h a t a d j u s t s the step s i z e ,  Aw  weight  complex  polynomials.  i n (2.11), the denominator  positive  the  It  <  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  that, (2.19)  2 min{l/(q q)} T  S Despite  i t s similarity  s u p e r i o r due  t o t h e use  t o the G r a d i e n t Method, t h i s a l g o r i t h m i s of o r t h o n o r m a l  with  the G r a d i e n t Method,  point  i f one  adjustment points  Klett  thus  change i n the  speeding  the C e r e b e l l a r Model as  estimate  convergence.  i t was  Discrete  adjusted minimizing  We  at  other  call  this  o r i g i n a l l y proposed  [ 2 6 ] a s a m o d e l o f l e a r n i n g i n t h e mammalian  as  As  s e v e r a l i t e r a t i o n s are required at a  a minimal  i n the space,  derived  t e r m s i n q.  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 causes  algorithm  polynomial  cerebellum.  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 follows:  At each t r a i n i n g p o i n t  the  weights  to e l i m i n a t e e r r o r i n the estimate at the p o i n t the  change  i n the estimate  at  by  previous  be are  while  training  p o i n t s , r a t h e r than 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 a l l other  points in  t h e s p a c e a s was  11  done f o r ( 2 . 7 ) .  The  Lagrange  multiplier formula that at  technique  identical  i n place  t o that  the nthtraining n-1 N  =  1/nZ  DQ  =  I  t o obtain  of Interference  of the P matrix  c  D  c a n be u s e d  there  adjustment  Minimization,  except  D which n  i s used  i s a matrix  point; m  p.p.  a weight  1  T  =  [(n-1)D  N  _  1  +  P  n  P  n  (2.20)  ]/n  where,  We  call  (2.21)  this  algorithm,  with  Interference  in  which  E„  n  We  =  call  using D  n  ~  1  these  Minimization algorithms  Minimization,  polynomial  orthogonal  /  Pointwise  terms  Interference two v a r i a n t s  in  the  basis  Minimization. c a n be  As  implemented  vector  are  made  the matrix, (2.22)  2  variants,  Pointwise  and the Pointwise  a r e summarized  Orthogonal  Cerebellar  i n table 2 . 1 .  12  Model.  Interference The  various  METHOD  1  2  NAME  Gradient  ESTIMATE  Method  . Learning Identification  3  Cerebellar  6  Pointwise Interference Minimization  — — Aw=uAfp  _Tf=w p  Aw=  -  A  P  f  T  f=w  Aw=-—• j— p P p  p  -Tf=w q  Aw=  ^  M  Table  2.4  2.1  CHOICE In  f=w  q  Aw=uAfq  f=w p  p D  -Tf=w q  conditions  fundamental  to  the  -,T— f=w q  normalize allows  properties occurring  variable  ranges.  confined  to  a  n  In  finite  of  due  to  p p '  T  1  r  q=E p, /-» — 1 7 n ^ ' n  q q T  _ ^  COMPARISON  learning  to  = E  r  E =D  17  n  n  _r>  -1/2 n  1  _ nP  the  a  characteristics  algorithms  input values  13  ALGORITHMS  algorithms,  particular  of  OF  convenient  v a r i a b l e s . Comparison  see  practice, range  n-1 2 /=0  algorithms  a l l input one  =l/n  -1p  — _ Aw=uAfq  learning  various  p  ^  M  At..  C O N D I T I O N S FOR  TEST  such  anomalies  Aw=  -TZ'-lZ  Model  1  D c  OF  is  _ n  T  of  step  /  q=Qp  Aw=-  T  Summary  first  -1/2 Q=P  q q  Model  comparing  dS  T  Pointwise Orthog. Interference Minimizat ion Pointwise Cerebellar  / pp S  q=Qp,  AfD  8  P=  =r  T  Orthogonal Interference Minimization  5  -,T— f=w p  NOTES  p p  Interference Minimization  4  WEIGHT A D J U S T M E N T  rather  combinations variables and  thus  that  are  than of  are  there  under  just input  normally is  not  a  /  <•  loss to  of  generality  allow  inputs  normalizing  p o s i t i v e and  were  -1  in  <  z  normalized <  /  s,  and  were  the  polynomial  function  to  We  =  these  1 +  z  found  +  1  z  that  randomly  1.  For  Figure  was  of  function  were  five  coefficients s=3  and  The follows:  to  to  assumed  +  be  by  for  trials  a l l 1's  v=3,  a  that  the  and  convergence  100  such  was  training test  v  the  to  with  s=2  and  be a l l  v=2,  the  was,  +  2  z z 1  of  (2.24)  2  convergence of  results  polynomials  when  estimating  when  with  estimating coefficients  the  algorithms,  where  for  of the  the  results  other  the  a  deemed  less to  points.  for convergence  have 100 in  14  the  learning point  training 0.01  the  has  cases  and  target  where  was  found  then  the  the  first  proven  considered.  to  are  results.  sequences was  the  shown  similar  1. the  K-G  throughout at  -1  trials  points  occurred  points  the  trials  showed  training  than  of  five  A l l of  various  at  of  cases  successive was  learning  coefficients  chosen.  several  error  and case  learned  chosen  each  with  100  the  a l l cases  between  e r r o r , Af, at  in  s  was  system  distribution  when  0.01  to  In  of  uniform  rule  the  z  K-G  randomly  but  v.  learned  representative  from  were  be  rates  stopping  than  adequate  are  compares,  results  a l l  2  +  2  described  2.1  represented,  combinations  variables,  example,  relative  generated  various  corresponding  estimate  functions  functions  less  was  2  have  for  for  input  estimate  set  whose  of  * f  be  order  that,  made  number  whose  coefficients  q u a n t i t i e s to  In  (2.23)  function K-G  such  variables.  1  Comparisons order,  negative  a l l input  to  i t  as be was  space of  the  be  an  Several  8 CASE s = 3 v = 3  h  METHOD 1 METHOD 2 METHOD 5 METHOD 3 I METHOD 6  RANDOM COEF'S  • ALL  Too" TRIAL NUMBER 0  polynomial  w i t h randomly chosen c o e f f i c i e n t s v e r s u s convergenc target  function  is a  c o e f f i c i e n t s , f o r case s=3 and v=3  15  target  polynomial  function  t.oo  Convergence  when  when  Tio"  F i g u r e 2.1  rates  rates  Too"  rs  with  is  1's  a  trials  were  acceptance  duplicated was  using a similar  r e q u i r e d a t more t h a n  points.  Convergence o c c u r r e d a f t e r  before.  The  uniform  distributions  training  a d j u s t m e n t was For the  the  those  f a c t o r u, u was  Method,  ( 2 . 1 3 ) was  the  satisfied  u = 2/m  successive  where  training  same number o f p o i n t s  successively  chosen  i n p u t v a r i a b l e s and  learning algorithms  causing divergence  Gradient  rule  one  as  from weight  performed at each p o i n t .  step size  without  of  100  the  p o i n t s were  stopping  that require s p e c i f i c a t i o n  c h o s e n t o be a s  to b e p o s s i b l e .  condition for  by  l a r g e as  allowable  For Method  non-divergence  of  1,  the  given  in  choosing,  .  (2.25)  For  M e t h o d 5, the C e r e b e l l a r Model, s u c h an e x p l i c i t f o r m u l a * for u was n o t o b t a i n e d ; f o r e a c h 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 w h e r e , u = 2 min{l/(q q)} = 2 min{l/(p Q Qp)} (2.26) S S T  T  T  * In  a l l  cases  u  was  obtained  by  generated  input v a r i a b l e vectors  obtained  a r e t a b u l a t e d i n t a b l e 2.2.  cases  that  F i g u r e 2.2  v=1.  cases cases  {z^.,,2  }.  I t was  shows t h e number o f s t e p s  The  except  cases,  s=1  c h o i c e o f u g i v e n by f o r s=3  and  v=3.  and  I t was  p o s s i b l e t o s p e e d c o n v e r g e n c e by 16  randomly  u factors  verified for  v=3,  m,  noted  s=3  and  Method  v=3,  near o p t i m a l f o r t h i s and  t h a t i t was  increasing u slightly  so  in several  r e q u i r e d f o r convergence  ( 2 . 2 6 ) was  w i t h a l a r g e number o f t e r m s ,  2000  The  such a c h o i c e of u i s near o p t i m a l  a f u n c t i o n of u f o r the and  considering  5. as s=3  in a l l other  typically from t h a t ,  i 8  •  GAIN FACTOR  F i g u r e 2.2  Convergence  Model,  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,  as  rates  for  v=3, and s=3 and v=1  17  Method  5,  the  Cerebellar s=3  and  given  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  choice  of  u t h e 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  space.  Thus i f o c c a s i o n a l g r o u p s o f s e v e r a l s u c c e s s i v e  points  were  algorithm  in  divergent  would d i v e r g e .  r e g i o n s of  the  T h i s was c o n f i r m e d  space,  the  training  the  by s e v e r a l  a  total trials  involving  10 o r more a d j u s t m e n t s p e r t r a i n i n g p o i n t i n w h i c h u s e  of  g r e a t e r than  a  u  that specified  by ( 2 . 2 6 )  would  lead  to  t o c o n s i d e r u as given  by  * divergence.  Thus i t seems r e a s o n a b l e  ( 2 . 2 6 ) t o be t h e b e s t c h o i c e when u s i n g M e t h o d 5, t h e C e r e b e l l a r Model.  s  6  .082  .022  t e m  5  .112  .036  .015  4  . 161  .064  .034  0  3  .251  .131  .093  .109  .147  —  2  .455  .333  .327  .448  .676  1 .08  1 .68  3.00  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  y 5  r d e r  —  Number o f V a r i a b l e s  Table  2.2  Step s i z e  factors,  u,  f o r M e t h o d 5,  the Cerebellar  Model  2.5 COMPARISON OF THE LEARNING Comparisons table  2.1  except  ALGORITHMS  were made o f a l l o f t h e l e a r n i n g a l g o r i t h m s i n m e t h o d s 4,  7 a n d 8.  18  Methods 4  and  7  are  equivalent same  t o m e t h o d s 3 and  number o f s t e p s .  completeness  only;  6,  r e s p e c t i v e l y , and c o n v e r g e  M e t h o d 8 was  included  in actual application,  in table  ensure  learning  n o n - d i v e r g e n c e d e p e n d s on E takes  place.  b e f o r e h a n d f o r method Figures for  2.3  v, ranged  for single trials  Figure  2.9  u c a n n o t be  of  Af,  to  of  changing  u as  calculated  required  The  number o f  1 t o 8 and t h e s y s t e m o r d e r , data  except f o r the f i r s t  the average of t h r e e  order  cases  trials.  f o r t h e c a s e s=3  learning algorithms. averaged  over  The  intervals  s,  represent  shows t h e 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 ,  various  magnitude  is  to converge.  a f u n c t i o n o f t h e number o f i t e r a t i o n s the  which  choice  show t h e number o f s t e p s  from  where t h e r e s u l t s r e p r e s e n t  using  The  1 t o 6 i n t h e c a s e s i n v e s t i g a t e d . The  results  for  8.  the v a r i o u s l e a r n i n g algorithms  ranged from the  Thus t h e o p t i m a l  t h r o u g h 2.8  input v a r i a b l e s ,  n  2.1  method 8 a p p e a r s  r e q u i r e t o o much c o m p u t a t i o n t o be p r a c t i c a l . to  i n the  graph of  100  there  is  M,  as  and  v=3  shows  the  training  iterations. For  cases  difference  having  between  a low s y s t e m o r d e r  most  of  d i f f e r e n c e s become p r o n o u n c e d considered. poorest  Method  algorithm.  1,  the  o n l y when h i g h e r  the Gradient  Method  various  3,  and  6th  approximately  order  systems  Interference  9, 20, 24 a n d  Identification,  i s similar  the  much  algorithms;  the  order  are  systems  Method, i s c o n s i s t e n t l y the  c o n s i s t e n t l y much b e t t e r t h a n t h e G r a d i e n t 5th  not  factor  Minimization,  Method; f o r 3 r d , 4 t h , of  improvement  28, r e s p e c t i v e l y . M e t h o d 2, i n performance  19  is  t o the Gradient  is  Learning Method  8 h  8  FIRST  ORDER  METHOD K METHOD • METHOD •  METHOD  CASES  6 5 3 2 1  • METHOD • NUMBER OF  3.00  4.00 8.00 NUMBER OF V A R I A B L E S  F i g u r e 2.3  Convergence  7700  8.00  r a t e s f o r v a r i o u s m e t h o d s when s=1  20  SECOND  ORDER  CASES  81 • 1  co8J  s I ID Z  8 •V.00  METHOD K METHOD • METHOO METHOD METHOD NUMBER  Too"  F i g u r e 2.4  17oo  4.00  8.00  NUMBER OF V A R I A B L E S  8.00  6 5 3 2 1 OF TERMS  7.00  Too  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 m e t h o d s when s=2  21  T H I R D ORDER  CASES  8" I  METHOD METHOD METHOD METHOD METHOD NUMBER  8  8 •f^o  i.oo  g u r e 2.5  i!oo  s.oo  a'.oo  Too  BTOO  NUMBER OF VARIABLES  Convergence r a t e s  6.00  6 5 3 2 1 OF TERMS  7.00  7. oo  f o r v a r i o u s m e t h o d s when s=3  FOURTH ORDER CASES  METHOD X METHOD • METHOD METHOD METHOD NUMBER 8.00  4.00  8.00  NUMBER OF V A R I A B L E S  F i g u r e 2.6  Too"  6 S 3 2 1 OF TERMS  ToT  6.00  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 m e t h o d s when s=4  23  F I F T H ORDER C A S E S  METHOD 6 METHOD 5 METHOD 3 METHOD 2 METHOD 1 NUMBER OF TERMS ~r  BTOO"  .rates  for various  "NUMBER OF VARIABLES  Figure  2.7  Convergence  24  l.oo  7.00  methods  when  7. oo  s=5  SIXTH  ORDER  CASES  METHOD X METHOD • METHOD METHOD METHOD NUMBER  iToo  iToo BTOO NUMBER OF VARIABLES  F i g u r e 2.8  To7  6 5 3 2 1 OF TERMS  Tib"  7.00  Convergence r a t e s f o r v a r i o u s methods when s=6  25  (L60  o ^0.00  F i g u r e 2.9  -103  iilo ITio 2~*0 TRAINING POINTS  Reduction of e s t i m a t i o n e r r o r ,  i.oo  A f , as 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 o i n t s f o r the case s=3 and v«=3 u s i n g t h e various iteration  methods  (estimation  error  interval)  26  averaged  over  each  100  i n c a s e s where t h e  system order,  s,  input  v,  Learning  variables,  b e t t e r when v>s, the  however,  Model, and  is' intermediate  Gradient  Method  Learning  Identification;  is  better  Pointwise  Interference  for  than  cases  except  t h a t when s>v  Learning  where  v=1  to  the  Method  Pointwise  is  similar  cases  v>1  i t i s b e t t e r by a f a c t o r  of  of  use  like  Cerebellum  is consistently  the  5, the  much  Identification.  Minimization where  i s much  i n performance  Minimization,  Interference Minimization,  method;  Identification  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  Cerebellar  Model  i s l a r g e r t h a n t h e number of  6,  the  best  Interference  to Interference Minimization  and  for  approximately  2,  e v e n when s=1 . The  choice  complexity  of  performance. performance  which  implementation Method  second  M i n i m i z a t i o n . As  3, only  will  be  as  to well  Interference t o Method  6,  depends  as  the  implementation.  i s not  Pointwise  as  each  training  better thus  D  the m a t r i x  R  must be  point.  Pointwise  Interference  however,  certain  Interference section  does  not  warrant  situations  Minimization  in  by  27  other  algorithms D  1  at  Minimization  is  o n l y a f a c t o r of 2  and  consideration. which the  i s appropriate.  2.7.  of  Minimization,  inverted yielding  than Interference M i n i m i z a t i o n typically  Interference  i t m i g h t seem, t h u s  Interference  u p d a t e d and  offers  the complexity  h o w e v e r , e n t a i l s much more c o m p u t a t i o n t h a n t h e as  the  achievable  Pointwise  as g r e a t  on  Minimization,  shown i n s e c t i o n 2.6,  Interference Minimization simplifying  algorithm  There  use  of  This  is  are,  Pointwise shown  in  2.6 IMPLEMENTATION CONSIDERATIONS FOR INTERFERENCE MINIMIZATION Interference Minimization achieves over  the  Gradient  i t s improved performance  Method and L e a r n i n g  Identification  at  the  e x p e n s e o f a more c o m p l e x 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 great as i t appears a t f i r s t .  variables (2.9)  are normalized  becomes  sparseness required  sparse.  a s i n (2.23), P  t o be e a s i l y  1  then  has a s p e c i a l  utilized  I f the  the matrix P  input in  1  form t h a t a l l o w s i t s  t o s p e e d up  the  computations  i n Interference Minimization.  Consider  the elements of t h e matrix P.  The e l e m e n t P.. i s i J  g i v e n by t h e i n t e g r a l , P.. = J p . ( z ) p . ( z ) 5S S  J  (2.27)  1  i n (2.2),  From t h e d e f i n i t i o n o f P/ (z) g i v e n C  (2.3) a n d (2.4) we  have,  1 p..  e  At  that  least  i  (e^ j j +e  c  half  the  variable,  z^,  polynomials  then  o f t h e e l e m e n t s P. . a r e z e r o a n d f o r v>>s are zero.  An  examination  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 .  the s p e c i a l has  v  ^)mod2=1 f o r any  t h e v a s t m a j o r i t y o f t h e e l e m e n t s P^. of  (2.28)  e  -1 -1 *=0 *  i t occurs  P. .=0.  , . n z . * / * y * ' 5z,..6z  =  l J  If  1 v  i n the set {p^(z)} are  banded n a t u r e  of P w i l l  ordered  If  appropriately,  become a p p a r e n t .  The m a t r i x  a " k i t e " form i n which a l l t h e nonzero elements appear as a  series  of  Operations  smaller  symmetrical  matrices  along  the  diagonal.  w i t h P c a n t h u s be done a s s m a l l e r t a s k s p e r f o r m e d on  the i n d i v i d u a l  smaller matrices. 28  T h i s means t h a t P  1  a n d Q have  t h e same " k i t e "  f o r m a s P.  o f P -1 a n d { p , ( z ) }  format  As a n e x a m p l e , f i g u r e  .-1 + - +  1 1 1 Z  z z 2  - -0  2  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  3 3 Z 2  Z  T  Z  1 1 1  Z  1 1 2 2 2 2 3 3 3 3  Z Z Z Z Z Z Z Z Z  Z  2 Z  Z Z Z  Z Z Z  Z  2 2 3 3 Z  Z  1 1 2 2 3 3 Z  Z  Z  1 1 2 2 3 3 Z  Z  Z  1 3 2 3 z z z Z  Z  s=3 a n d v = 3 .  f o r the case  {p*(i>}  Z  2.10 shows t h e  Z  Z  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0 0 0 - + 0 0 0 + 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  - -+ --  0 0 0 0  -+ -0  0 0 0 0 0 0 0 0 0 0 0  0 0 0 0 + 0 0 0 0 0 0 0 0 0 0 0 0  0 0 0 0  0 0 0 0 - 00 -+ 00 0 + 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0  0 0 0 0 0 0 0  0 0 0 0 0 0 0  0 0 0 0 0 0 0  +  - -+ -- + -0 -0 0  0 0 0 0 0 0 0 0 0 0 0 0 + 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 -  -+ - - + -+ -0 -0 0  0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 o f z e r o a n d n o n - z e r o e l e m e n t s o f m a t r i x P -1  f o r t h e case  s=3 a n d v=3 (+ = p o s i t i v e ,  An found. The  - = negative, 0 = zero)  e m p i r i c a l method of d e s c r i b i n g t h e form Consider  a system of o r d e r  s,  of P  1  having v input  been  variables.  number o f t e r m s i n { p ^ ( z ) } a n d h e n c e t h e s i z e o f P i s , (v+s)! m(v,s) =  It  has  is  (2.29)  v!s!  useful  t o define the value of t h i s  function  when  arguments a r e z e r o o r n e g a t i v e ; m(-v,s) = m(v,-s) = 0 m(0,s)  = m(s,0)  (2.30)  =1  (2.31)  29  the  for  v,s  positive.  matrices  forming  P  The  s i z e and m u l t i p l i c i t y  c a n t h e n be o b t a i n e d  1  of  the  smaller  from t h e terms of  the  identity, m(v,s) =  s\2+1 Z i =1  where \ r e p r e s e n t s  In  1  (2.32)  1  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  (2.34)  /  = m(v,/-1)  the matrix P ,  along  r.d.  1  the  t h e number o f d i f f e r e n t  diagonal corresponds of (2.32).  The  o f e a c h i s g i v e n by t h e f a c t o r  occurrences  T h e r e a r e s\2+1  nz(v,s)  For  s\2+1 = Z i = 1  growing  that of m(v,s). The  d^ a n d t h e  matrix  P  of  sizes.  number  The t o t a l  of  number  2 s,  nz(v,s)  i s much s m a l l e r t h a n  m(v,s)  i n s i z e a s v i s i n c r e a s e d i n manner more  the  reduces  when i m p l e m e n t i n g  the  (2.35)  t h u s becomes a l m o s t  1  in  i s ,  T h i s i s shown f o r t h e c a s e  Normalization significantly  r^. .  , r.d.  a system of order  f o r v>>s,  1  submatrices  unique submatrix  o f e a c h i s g i v e n by t h e f a c t o r  of n o n z e r o e l e m e n t s i n P  of  t o t h e number o f t e r m s  summation size  sizes  input  s=3 i n f i g u r e  d i a g o n a l i n form f o r  variables  t h e number o f  as  (2.23)  multiplications  Interference Minimization.  30  in  like 2.11.  v>>s. thus  required  F i g u r e 2.11  Total  number  of  elements and order of m a t r i x P  elements, 1  number  f o r c a s e s where s=3  31  of  non-ze  2.7  APPLICATIONS  WHERE P O I N T W I S E  INTERFERENCE  MINIMIZATION  IS  APPROPRIATE In at  a  in  the  was to  the  derivation  training  point  estimate  assumed allow  a  at  for  of  was  eliminated while  other  these  Interference Minimization,  points  other  non-uniform  i n the  points.  weighting  when  accuracy  i s more  than  others.  With  critical  this  more  [26], at  minimizing  space. A  more This  certain  general  the  An  the  equal  general would  approach  change  weighting  approach  be  regions  error  is  appropriate  of  the  space  equation  (2.7)  becomes, c  where  =  h(S)(Aw p) T  h(S)  optimum the  J S  is  weight  definition P  =  J S  One  input  and  one  the  of  the  variables  as  appropriate effectively  positive  formula P  (2.36)  weighting  remains  as  function.  before  to  demands  have  a  If then  distribution use  the  that  except  most  probability h(S) P  i s an  i s sparse  h(S)  i s not  In  m u l t i p l e of  weighting  probability  estimate  are  Pointwise  learns h(S).  non-uniform  non-uniform  improve  space  S  The that  becomes,  that  h(S).  scaler  Af)  (2.37)  the  to  -  T  matrix  use  forming  the  a  to  a(Aw p  8S  T  variables  variables  be  of  h(S)pp  appropriate  to  adjustment  wishes  regions  +  strictly  application  the  If  a  5S  2  most  as  distribution  rapidly  probable.  distribution even  i s when  in  It  is the  input  f u n c t i o n of  the  input  described previously.  known  the  32  limit,  then  of  beforehand  then  Interference Minimization  the  the  the  matrix  corresponding  D  n  matrix  i t is as  i t  converges P.  For  example,  if  the  input v a r i a b l e s are  n o r m a l i z e d as  in  (2.23)  then, D  = P/2  (2.38)  00  Pointwise with  2.8  the  Interference Minimization l e a r n i n g of  the  combines the  l e a r n i n g of  P  estimate.  SUMMARY A  their  family  of l e a r n i n g a l g o r i t h m s has  convergence  Interference either  characteristics  Minimization  has  of  the  Pointwise  Interference  the  best a l g o r i t h m  on  the  improvement  over  would not  The  been f o u n d t o  be  Pointwise  One  variables  method  of  superior  to  situation  Minimization rapid  significant w h e r e i t may  Interference Minimization  has  been found t o  convergence,  t h a t a r e most p r o b a b l e and  effectively  unknown. learns this  the  be  however, small  in  appropriate  i s when one  and  algorithm to  wishes to input  be  use have  variable  the p r o b a b i l i t y  distribution,  Interference  Minimization  Pointwise distribution.  33  is  increase  r a p i d c o n v e r g e n c e i n t h o s e r e g i o n s of t h e  is  Further,  simplifies  Interference Minimization the  and  Minimization.  the b a s i s of  seem t o j u s t i f y  complexity.  h(S),  compared.  input  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  space  introduced  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 .  normalization  most  been  S E L F - L E A R N E D  3  One  of  C O N T R O L  O F  A  TWO  L I N K  t h e 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 ,  l e a r n i n g a l g o r i t h m s c a n be u s e d t o a c h i e v e of  a manipulator.  of  p o l y n o m i a l terms c o u l d model t h e  functions  that  manipulator, methods  the  and  that  be  the  self-learned  multi-variate,  dynamics  Interference  used t o l e a r n  and  RESOLVED  that  control  Minimization  the  to achieve  ACCELERATION  non-linear  kinematics  weighting  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 ,  functions necessary  3.1  is  I t h a d 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  describe  could  without  M A N I P U L A T O R  or  of  a  related  coefficients, and t h u s  learn  control.  CONTROL  WITH  CARTESIAN  PATH  SPECIFICATION G i v e n t h a t one h a s t h e a b i l i t y non-linear necessary  functions  this capability with a  A  family  o f s u i t a b l e methods  One  manipulator  Inverse Plant,  multi-variate,  manipulator,  utilize  called variously, etc.  a  to  technique.  ,  describing  to learn the  i t  suitable  we  manipulator In by  then  control  [20,29,30] a r e  those  I n v e r s e System, I n v e r s e Problem  e x a m p l e o f t h e s e m e t h o d s t h a t h a s been a p p l i e d  to  control  as  is  Resolved  Acceleration  Control  d e s c r i b e d by L u h 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 that  is  propose  to  apply  learning  algorithms  framework to  learn  control.  general,  the dynamics of a manipulator  a group of f u n c t i o n a l  c a n be d e s c r i b e d  r e l a t i o n s h i p s t h a t we c a l l  dynamics;  34  the  direct  a  dir.dyn(7,a,a)  (3.1)  a  / a 6t  (3.2)  a  / a 6t  (3.3)  where  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  g e n e r a l i z e d torques or f o r c e s , to in  position  a  and  acceleration  a.  The  as a p p r o p r i a t e , t h a t a r e  moving exact  with velocity  a  will  of  applied  as  to  is  result  in  form of the f u n c t i o n s c o m p r i s i n g  d y n a m i c s r e l a t i o n s h i p s a r e d e p e n d e n t on t h e  configuration lengths,  i s a vector  A p p l i c a t i o n o f t o r q u e 7 when t h e m a n i p u l a t o r  the j o i n t s .  direct  a n g l e s and 7  t y p e and  number o f l i n k s ,  l i n k m a s s e s and mass d i s t r i b u t i o n s ,  the  manipulator  and  joint  the  link  friction  and  gravity. The  manipulator  dynamics  which  can  are  r e l a t i o n s h i p s t h a t we  be  controlled  described call  by  the i n v e r s e  a  using group  its of  inverse functional  dynamics;  7 = inv.dyn(a,a,a)  (3.4)  where, a = 6a(t)  (3.5)  a = 6a(t)  (3.6)  8t  6t  To  achieve  position  acceleration  a, t o r q u e 7 must be a p p l i e d , a a n d  differentiating not easy  a when m o v i n g a t v e l o c i t y  the path s p e c i f i c a t i o n ,  to o b t a i n these r e l a t i o n s h i p s ,  m a n i p u l a t o r s having s e v e r a l degrees If  of  and  a c a n be o b t a i n e d  a(t).  in be  In g e n e r a l i t i s  especially  for  complex  freedom.  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 ,  35  a  i t would  appear  to  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  Given to  t h a t one  calculate  relationships, manipulator expressed  torque apply  to  loop  control.  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  the a c t u a l p o s i t i o n ,  can  by open  t h e n g i v e n t h e d e s i r e d p a t h o.^(t) 7(t)  r(t)  move  equal  using  to  inverse  dynamics  t h e m a n i p u l a t o r and  such t h a t  m a t h e m a t i c a l l y ' as  the  a(t)  equals  a definition  of  one  cause  a^(t). the  the  This direct  is and  i n v e r s e dynamics, namely, a(t) given  = dir.dyn(inv.dyn(a^,a^,a^),a,a) = a^(t)  (3.7)  that, a(t )  = a (t )  (3.8)  a(t )  = a (t )  (3.9)  Q  d  Q  and, Q  In  Q  p r a c t i c e such a c o n t r o l technique  precision the  d  in calculations,  i s poor  due  to  finite  i m p r e c i s e knowledge of parameters  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,  lengths,  e t c . and  the environment. inverse  dynamics  disturbances,  relationships  and  tracking errors w i l l  minimal  control  7 = inv.dyn(a j+k (  Application  of  1  f o r p o s i t i o n and  and  cause  velocity  ( a ^ - a ) + k j ( a ^ - a ) ,a,a)  to  adjust  errors; (3.10)  such torques r e s u l t s i n m a n i p u l a t o r  motion  follows, a -  dir.dyn(inv.dyn(a +k (a -a)+k2(a -a),a,a),a,a)(3.11) d  1  d  36  the  control.  c a n be a c h i e v e d by u s i n g f e e d b a c k  applied torques to correct  of  by  environmental  tend to accumulate  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 Good  joint  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 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  in  d  as  which  by  the  definition  of the d i r e c t  and  inverse  dynamics  yields, a = a^+k ( a ^ - a ) + k ( a j - a ) 1  2  (3.12)  (  0 = (a ~a)+k (a -a)+k (a -a) d  Defining e  1  d  = a, - a a  a  in  equation  (3.14)  a second  with  0 = e  + k,e 1 a  a  order,  stability,  This  1  is  change  o r e q u i v a l e n t l y t o have e* d e c r e a s e a  and f o r a c r i t i c a l l y  with  time,  damped r e s p o n s e  one  =4k . 2  f o r manipulator  c o n t r o l has  already  by L u h e t a l [ 3 0 ] f o r a c o m p l e x m a n i p u l a t o r  to  in  (3.15)  0  d y n a m i c s were known a n a l y t i c a l l y . work  differential  + k e 2 a  technique  demonstrated  homogeneous  time,  must c h o o s e k ^ O , 2  must c h o o s e k  linear,  constant c o e f f i c i e n t s d e s c r i b i n g the  position error with  one  (3.13)  d  the p o s i t i o n error as,  results  For  2  determine  been whose  One o f t h e m a i n g o a l s o f t h i s  whether t h i s c o n t r o l  technique  can  be  s u c c e s s f u l l y a p p l i e d when 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 a r e learned  without  recourse  t o a n a l y s i s of  the  dynamics  of  a  manipulator. In al, it  resolved acceleration  the path s p e c i f i c a t i o n  c o n t r o l a s p u t f o r w a r d by L u h e t  i s i n joint coordinates.  Typically,  w o u l d be more d e s i r a b l e t o s p e c i f y t h e m a n i p u l a t o r  another hand.  c o o r d i n a t e system For  manipulator  example, end  which  i s more s u i t a b l e  path  f o r the task at  one may d e s i r e t o s p e c i f y t h e p a t h o f  point i n Cartesian  s p e c i f y the path of the manipulator 37  coordinates, joints  in  rather  the than  i n joint coordinates.  Manipulator be  achieved  using  conjunction inverse  end p o i n t  with  control i n Cartesian  the inverse  kinematics  coordinates  can  of t h e m a n i p u l a t o r  in  the i n v e r s e dynamics of the  kinematics  are  described  by  a  manipulator.  group  of  The  functional  r e l a t i o n s h i p s as f o l l o w s : a - pos.inv.kin(x)  (3.16)  a = vel.inv.kin(x,a)  (3.17)  a = acc.inv.kin(x,a,a)  (3.18)  Given  the  point  vectors  position,  coordinates,  x and x d e s c r i b i n g t h e  velocity  one c a n o b t a i n  the m a n i p u l a t o r  and  acceleration  the vectors  at a given  unique.  I t i s thus necessary t o apply  inverse  kinematics  example,  a  t o be d e s c r i b e d  revolute  because  constructed; full  circle For  of  describing  p o s i t i o n i s not  c o n s t r a i n t s to permit the  by f u n c t i o n a l r e l a t i o n s h i p s . to  Such c o n s t r a i n t s o f t e n  manner  place  in  which  revolve  physically  manipulators  joints often are r e s t r i c t e d  are  t o l e s s than a  of r o t a t i o n .  typically  the  manipulator.  the  the  revolute  one  orientation  a and a  Cartesian  m a n i p u l a t o r s h a v i n g more t h a n t h r e e  manipulator,  Cartesian  j o i n t may be c o n s t r a i n e d  t h r o u g h l e s s t h a n it r a d i a n s . exist  a,  end  in  the combination of j o i n t a n g l e s t h a t w i l l  m a n i p u l a t o r end p o i n t  For  manipulator  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 .  Typically, the  x,  degrees of freedom,  w a n t s t o c o n t r o l more t h a n j u s t t h e e n d p o i n t  one  For  example,  can  specify  of the l a s t  hand o r t o o l .  link  with a s i x degree the  end  point  which i s t y p i c a l l y  In t h i s case,  38  of  path thought  of  freedom and of  the as  the c o n s t r a i n t s d e s c r i b i n g the  orientation inverse can  of  the  hand,  which are necessary to  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  be v i e w e d a s p a r t  permit  the  relationships,  of t h e 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  can  be  that  we  constraints. The  inverse  combined call  to  kinematics  and t h e i n v e r s e  form a group of f u n c t i o n a l  the Cartesian  inverse  dynamics  relationships  dynamics;  7 = cart.inv.dyn(x,x,x)  (3.19)  where, £ = 6x(t) 6t  (3.20)  x = 6x(t) 6t  (3.21)  Similarly, combined call  t h e d i r e c t k i n e m a t i c s and t h e d i r e c t dynamics can to  form a group of f u n c t i o n a l  the Cartesian  relationships  that  be we  d i r e c t dynamics;  x = cart.dir.dyn(7,x,x)  (3.22)  where,  The  x = J x 6t  (3.23)  x = / x 5t  (3.24)  direct  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 a r e  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  described  by  follows,  x = pos.dir.kin(a)  (3.25)  x = vel.dir.kin(a,a)  (3.26)  x = acc.dir.kin(a,a,a)  (3.27)  Given  t h e v e c t o r s a,  positions, vectors  x,  a and a d e s c r i b i n g  velocities x  and  x  and  the manipulator  accelerations,  describing  39  a  the  one c a n o b t a i n  manipulator  end  joint the point  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 . In  a manner a n a l a g o u s 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  control along  the  with  manipulator  using the C a r t e s i a n  f e e d b a c k t o c o r r e c t p o s i t i o n and  T = cart.inv.dyn(x j+k (  x=  1  inverse  can  dynamics  velocity errors;  ( x ^ - x ) + k ( x ^ - x ) ,x,x)  (3.28)  2  cart.dir.dyn(cart.inv.dyn(x^+k (x^-x)+k (x^-x),x,x),x,x) 1  2  (3.29) x = x  + ( 3  k  (x -x) + k (x ~x)  1  d  2  (3.30)  d  0 = (x -x)+k (x -x)+k (x ~x) d  Defining e one  can  critical  2  (3.31)  d  e r r o r as,  ~ x  (3.32) and  1  k  2  as b e f o r e  to  achieve  Cartesian than  the  coordinate  kinematics.  Such  inverse  dynamics  cylindrical  i n v e r s e arm  r e l a t i o n s h i p s due  transformation through would  be  or s p h e r i c a l ,  the t a s k a t hand. respect  the  and  r e l a t i o n s h i p s are  use  t h e c a s e i f one  The  key  to a coordinate  is  acceleration  control  specification Cartesian When  thus  in  a  can  the  inverse  systems,  this  such  as  given  point i s that c o n t r o l i s desired with  s y s t e m t h a t may  to  of  inclusion  t h a t m i g h t be more c o n v e n i e n t  for d e s c r i b i n g manipulator work  t o the  more  w a n t e d t o use  approach i n conjunction with other coordinate  one  stability  -  d a m p i n g of C a r t e s i a n p o s i t i o n e r r o r s .  complex a  d  choose k  The  of  d  the C a r t e s i a n p o s i t i o n = x  x  1  dynamics.  determine be  not  be  t h e most n a t u r a l  A second major g o a l  whether  learned  successfully applied  task oriented coordinate  of  resolved using  system  path  such  as  coordinates. applying  resolved  acceleration  40  control  using  Cartesian  c o o r d i n a t e s f o r end  necessary  t o observe  to  use  feedback  Ideally, path. and  one  t h e end  to  c o u l d use  a c c e l e r a t i o n s and  can  be  much  simpler  evaluation  to  the d i r e c t end  d e r i v e than  requires  nothing  velocity  order errors.  the  The  velocities  direct  than  to  kinematics  [ 4 4 , 4 6 ] and  inverse  more  a n a l y s i s and  point  kinematic r e l a t i o n s h i p s  point path.  c o s i n e terms.  they  kinematics.  are Their  multiplication  and  N e v e r t h e l e s s , s u c h a method  time  consuming,  Interference  M i n i m i z a t i o n i t s h o u l d be  functional  relationships that describe  the  kinematics.  This  would  manipulator  without  the  r e s o r t t o a n a l y s i s and manipulator  3.2  in  is  manipulator-  calculations.  With  path  it  s y s t e m t o m e a s u r e t h e end  by a g e n e r a l t e c h n i q u e  requires manipulator  learn  and  can measure j o i n t p o s i t i o n s ,  use  s u m m a t i o n o f s i n e and  specific  position  a vision  the manipulator determined  specification,  point Cartesian position  correct  E q u i v a l e n t l y , one  determine  point path  end  allow use  Cartesian  of a v i s i o n  calculation  path system  of the d i r e c t  possible the  direct  control and  to  of  without  kinematics.  p o i n t c o u l d t h e n be o b t a i n e d f r o m s i m p l e  The  joint  measurements.  A TWO In  L I N K MANIPULATOR order to t e s t the  manipulator, vehicle. associated  a s i m p l e two  feasibility  of l e a r n e d c o n t r o l of chosen as a  test  manipulator  and  c o o r d i n a t e r e f e r e n c e f r a m e s a r e shown i n f i g u r e  3.1.  The  structure  l i n k manipulator of  the  41  two  was  a  link  3.1  Figure  Such  a  First,  A two l i n k  simple manipulator  was c h o s e n  for  two  basic  reasons:  i t s d y n a m i c s and k i n e m a t i c s a r e known i n a n a l y t i c a l  This permits advantages  that  observation mistakes  against learned to  simulation brings;  exact of  specification  a l l variables,  a r e made.  analytical  Also,  dynamics  which  and  no t i m e of  a n d no  parameters, physical  kinematics can  one c a n compare c o n t r o l  be s i m u l a t e d ,  consuming  c o n t r o l performance  dynamics and k i n e m a t i c s .  inordinate  form.  t h e arm t o be s i m u l a t e d on a c o m p u t e r , w i t h a l l t h e  development,  was  manipulator  Secondly,  time  42  when as  as the to  accurate  damage  performances  i t had t o be s i m p l e  amounts o f computer  serve  hardware  avoid  when  using  the  benchmarks when  using  manipulator requiring  f o r accurate simulation  of  numerous  movements.  In s p i t e of the s i m p l i c i t y  o f t h e two l i n k  manipulator,  i t s kinematics  and dynamics a r e d e s c r i b e d  non-linear,  multi-variate  functions  complex m a n i p u l a t o r s . manipulator  will  more c o m p l e x The  These  of  be r e p r e s e n t a t i v e o f p o s s i b l e  angles,  basic trigonometry  v e l o c i t i e s and a n g u l a r  1 sin(a  x  1  =  =  l c o s ( a )d  1  +  x  2  =  •  1 sin(a )d  1  + l sin(a +a )(d  = 1 -  l c o s ( a )d  1  1  1  1  1  1  1  1  1  2  1  1  2  to  1  2  2  1  2  1  )d  1  1  obtain  the  i t is  2  joint  d  l )  ( 3 . 35)  1 + d  2 )  ( 3 . 36)  1 +  2  1  1  2  + l cos(a 2  inverse  '2 d  + a  2  a  1  +  1  + a  2  2  ) )(d +d ) 1  necessary t o apply  2  ( 3 . 37)  2  ( 3 . 38)  2  ) )(d +d ) 1  of  kinematics  relationships possible.  be b e t w e e n  1+  1  + l sin(a +a )(d ]  2  of  ( 3 . 34)  l cos(a +a )(d  + 1 cos(a )d  functional  i n terms  2  + ljcostc^+c^) (d 2 - 1 sin(a )d - l sin(a l  l sin(a  manipulator  Cartesian  - l ^ o s f a ) - l cos(a +a )  1  To  the  angles.  ( 3 . 33)  2 •  =  joint  to yield  be  Cartesian  ) + l^intc^+a^  X  2  can  a r e as f o l l o w s :  1  x  using  a c c e l e r a t i o n s . The d i r e c t  x  x  a  to c a l c u l a t e the  a c c e l e r a t i o n of t h e end p o i n t  angular  •  more  t h e two l i n k  results  t h e end p o i n t as a f u n c t i o n of t h e  and  kinematics  of  o f t h e two l i n k m a n i p u l a t o r  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  velocity  typical  We e x p e c t t h a t r e s u l t s u s i n g  kinematics  by u s i n g  position  is  manipulators.  direct  obtained  as  by h i g h l y  a  2  the  two  constraint  I n t h e f o l l o w i n g we  0 a n d 7r r a d i a n s .  The i n v e r s e  to  link make  restrict  kinematics  can  t h e n be w r i t t e n a s , a  2  = arccosUx^  + x  2 2  - 1^  43  - 1  2 2  )/21 1 ) 1  2  (3.39)  a  = arctan(x ,-x )  1  1  a r c t a n ( l  _  2  s i n ( a  2  2  ) +  l cos(a )) 2  2  (3.40) d  = (l sin(a +a )x  1  2  1  2  -  l  l cos(a +a )x )/l l sin(a ) 2  1  2  2  1  2  2  (3.41 ) d  = [(-l sin(a )x  2  1  1  + 1 cos(a )x )/l.l sin(a )] - d  1  1  1  2  2  2  1  (3.42) a  =  1  (l sin(a +a )x 2  1  + l  2 2  2  - l cos(a +a )x  1  2  1  2  1 l cos(a )d  +  2  1  2  2  (d +d ) )/l l sin(a )  (3.43)  2  1  2  1  2  2 1  2  2• 2  d  = [ ( - 1 s i n ( a . )x  2  1  to  that  cosines it  function  a value of a  i s useful  1  and  = (1  1  a r e used  t o use t h e f o l l o w i n g  sin(a ) cos(a )  1  2  -it and it.  that  = (x^ + x  2  2  (3.44)  2  h a s two a r g u m e n t s and  between  cos(a ) 2  1  2  1  arctan  yield  - 1^ d  l ^ c o s U ^ (d +d ) )/l l sin(a ) ] -  Note  + l^osfa^x^  1  - l ,  2 2  " cos(a ) ) 2  i t i s the s i n e s  i n subsequent  1  l  2  ) / 2 1 1  2  1  (3.45)  2  l / 2  x (1.+l cos(a ))]/(x  -  2  and  calculations,  2  = [x l sin(a )  assumed  formulas,  - 1  2  Since  i s thus  2  2  2  1  2 +x  2  (3.46) 2 ) (3.47)  sin(  t t l  )  = (1  The d e r i v a t i o n appendix The  - cos(  inverse  /  (3.48)  2  inverse  d y n a m i c s o f t h e two  defined  as the d i f f e r e n c e  The  l  kinematics  i s shown  in  A.  using  L  i  of t h e d i r e c t and  obtained  potential  ) ) 2  a  Lagrangian mechanics between  link  manipulator  can  [ 3 7 , 4 6 ] . The L a g r a n g i a n L i s t h e k i n e t i c e n e r g y K and  energy P of the system,  = K - P  inverse  (3.49)  dynamics e q u a t i o n s a r e o b t a i n e d a s ,  44  be  the  where  are the coordinates  energy  are expressed,  T.  the corresponding  a  torque,  to  as  These  2  upon  and  oriented  gravitational T  1  of  having  =  1 i  d  i  +  link  as  d  +  r  = d  2  2  a  1  d  + d  1  +  2  d  +  d  the  111 1 a  112 1 2 d  2  d  d  +  + d  2  2  and  d  +  d  a  and  force  angular  coordinates are  referred  1  and  1  d  1 ,  d  l21 2 1 d  d  +  2 1  d  221 2 1 d  d  link  2  3.1  l22 2  to  with  a  two  link  masses  respect  m  and  1  t o g,  the  inverse dynamics,  2  +  d  2  2  +  2  d  1 d  (  3  *  5  1  )  (  3  *  5  2  )  2 2  2  where, d  =  n  (m +m )l 1  + m  2  2  l  2  l  2  2m l l cos(a )  +  2  2  1  2  (3.53)  2  2 d  l  d  1 1  = m  2  1  =  0  l  2  + m l l cos(a )  2  2  1  2  l22  =  ~ 2 ^ " 1 •'•2  d  112  =  ~ 2 1 2 * ^ 2^  d  t21  =  d  m  l  m  1 2  =  1  21  =  d  22  =  d  211  =  d  m  -  (3.54)  2  (3.55)  d  1  i  1  s  2  S  n  i  n  ^ 2 ^  Sin  (3.56)  a  (3.57)  a  ^  a  2 ^  (3.58)  (m +m )gl sin(a ) + m gl sin(a +a ) 1  2  1  m  2  1  m  2 1 2 ^ ^ 2^ l  2  2  2  1  2  m  2  1  +  m  1 2  1  1 1  2  c  o  s  (  a  2  2  2  1  (3.59)  2  (3.60)  )  (  S  n  or  or  following  d  velocities,  linear  mechanics  +  d  n  212 1 2 d  2  is a  i  in figure  yields  potential  and c o o r d i n a t e s .  lengths  shown  l2 2  d  torques  and  i s either  a  whether  Lagrangian  field,  a  kinetic  torques.  torques  generalized forces,  manipulator m,  f o r c e s or  forces,  Application  the  d^. a r e t h e c o r r e s p o n d i n g  depending  coordinate.  i n which  3  ,  S  1  }  (3.62)  a  45  The  d  2  2  2  =  0  (3.63)  d  2  l  2  =  0  (3.64)  0  (3.65)  d  2 2 1  =  d  2  = m gl sin(a +a ) 2  2  1  derivation  of  the  (3.66)  2  inverse  and  follows  the approach used  the  results  differ  an  arithmetic  derivation  error  neglects  simplification, would  slightly  tend  friction, The  dynamics  by  from  [ d  a  2  =  [d  l  2  ( r  dynamics  (T  2 1  inverse  -  2  1  g ) 2  - g,)  point  o s c i l l a t i o n s that  can  =  This  from a c o n t r o l  direct  1  by  damped be  + d  n  ( g  2  and  -  T )]/A  -  T  t  also  that  is  this  necessarily  whereas  through  are as  a  with  no  system. algebraic  follows, (3.67)  Q  2  that  there  the control  obtained  dynamics  + d^tg,  by  as  B  of view. F r i c t i o n  occur  out  however,  Paul  i s not  forces.  be  a  given  frictional  o s c i l l a t i o n s must  the  those  Note,  Note  though,  of  [46].  i n appendix  i n the h i s d e r i v a t i o n .  t o damp a n y  manipulation  Paul  i s shown  ) ] / d  (3.68)  0  where, d  0  =  d  l2 21  "'  d  91  =  d  111 1  2  9  =  d  211 1  2  d  +  d  d  d  A  It  2  is  d  the  +  direct  ( 3 . 69)  11 22  d  122 2 , .2 222 2 d  +  2  d  1l2 1 2 d  d  +  +  a  dynamics  d  d  l21 2 1  +  221 2 1  +  d  in conjunction  d  d  d  with  d  d  1 2  the  ( 3 . 70) ( 3 . 71 ) integral  equations,  that  a  =  /  a  o  =  / a  allows  utilizing To  6t  (3.72)  8t  (3.73)  simulation  numerical achieve  of  the  integration  learned  manipulator  on  a  computer  routines.  resolved  46  acceleration  control  in  a  Cartesian dynamics or  reference  and the inverse  i t s equivalent  learn  frame  the  direct  kinematics.  f o r feedback kinematics.  simple  two l i n k  learned  are multi-variate  3.3  SIMULATION Testing  i t i s necessary  manipulator  OF T H E  control  of  the  performed  on  a - VAX-11/750  simulation  of  computing control shown  systems,  devices that system  The  [58].  continuously,  that  even  to  'for a  relationships  computer  such  would A  was c a r r i e d  manipulator.  ACSL  The  to  be  using  ACSL  permits  systems,  be u s e d  schematic  as  t o implement  (advanced  as well the  a  a was  the simulation  such  diagram  out using  simulation  as the manipulator,  time  blocks  physical  continuous  necessary  of as  digital  manipulator  of the simulation  i s  3.2.  various  separate  system  non-linear.  systems  language).  discrete  in figure  systems;  two l i n k  simulation time  the functional  inverse  a vision  i t i s  I t c a n be s e e n  the  MANIPULATOR  of various  continuous  To r e p l a c e  purposes,  and h i g h l y  simulation  continuous  to learn  time  entities. systems;  whereas  systems  in figure  whose  Derivative systems  discrete states  3.2 c a n b e  blocks  whose  of  at discrete  states  points  as  represent change  blocks represent discrete  change  47  thought  time  i n time.  useobs  "le  dlfferantlatton  Machine 2  differentiation  vtalon tyatea)  learning  =  *e K  a  CD  analytical direct dynanlca  analytical dlract kinematic* .  DERIVATIVE BLOCK ARM  taaeh  ^ ri OJ  DISCRETE BLOCK SAMPLE  learning Machine 1  taaeh learning Mchlna 2  DISCRETE BLOCK LEARNJ  DISCRETE BLOCK LEARN 2  DERIVATIVE  BLOCK  ARM  Derivative Given  b l o c k ARM r e p r e s e n t s  a statement of the d i r e c t  t h e two l i n k  manipulator.  dynamics i n d i f f e r e n t i a l  a = dir.dyn(7,a,a) the  integration  form, (3.74)  algorithms  o f ACSL a d v a n c e t h e s t a t e s  of  the  system, a and a, a s , a = /  a 6t  '  (3.75)  a = J a 5t Of  the  various  ACSL,  the  algorithm  was  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 e a c h i n t e g r a t i o n  step  Adams-Moulton chosen.  state  error  t o be l e s s t h a n  up t o t h a t p o i n t  about  1.0E-6.  achievable.  algorithms Kutta, time  t h e maximum v a l u e o f  i n t h e s i m u l a t i o n and t h e  Since the order  to  i t  absolute to  Other f i x e d  achieve  v a r i a b l e order  i s only appropriate  and  which  f i x e d order  such as E u l e r ,  comparable e r r o r  stepsize,  times  stepsize,  Runge-Kutta,  for  typically bounds.  o f what  integration  2nd o r d e r  Runge-  r e q u i r e d more The  Gear's  certain  difficult  CPU Stiff  [58].  integration  ACSL e n s u r e s  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 d a t e when a n y o t h e r b l o c k s a r e b e i n g  executed.  appears as a continuous  49  by  a l g o r i t h m was n o t c o n s i d e r e d  i s not the case here  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  be  of magnitude of the s t a t e s i s  e r r o r bounds a r e near t h e l i m i t  a v a i l a b l e i n ACSL,  problems, states  these  and 4 t h o r d e r  variable  the  1.0E-6 t i m e s  1 a n d 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 u s e d  ACSL f o r s t o r a g e ,  as  v a r i a b l e order  i n t r o d u c e d a t e a c h 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  l e s s than  is  integration algorithms available i n  variable stepsize,  was r e s t r i c t e d the  (3.76)  Thus  that at the  system t o  the r e s t of the s i m u l a t i o n . DERIVATIVE  BLOCK  Derivative current  block  PTHGEN  path s p e c i f i c a t i o n ,  time,  t . No  system  with  block  PTHGEN  integration states  i s used  specification  contains x^,  code  x^ and x^,  i sperformed  to  here.  i n t h i s manner s i m p l y t o e n s u r e whenever  another  the  given thecurrent  by PTHGEN a s  i snot being modelled  i s current  compute  a  A  dynamic  derivative  that  the  block  is  path being  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 t h e code i n each b l o c k uses  the path  specification.  P a t h s c a n be e i t h e r  linear  or  required  to  as described i n section 3 . 4 .  circular DERIVATIVE  BLOCK  Discrete  CALC  b l o c k CALC p e r f o r m s  the computations  control  themanipulator.  Inputs a r e the s p e c i f i c a t i o n  desired  Cartesian  x^,  resulting that  path,  joint path,  a,  x^ and x^,  a and a.  i s applied t o the manipulator.  block,  i t i sexecuted as d i s c r e t e  t  The o u t p u t t o r q u e  ,_.  and  and t h e view of t h e  The o u t p u t  7  discrete  i n t i m e a s f i x e d by t  The t i m e r e q u i r e d t o p e r f o r m  o f t h e CALC b l o c k w o u l d d e t e r m i n e  t , i n an a c t u a l c a xc  i s the torque  S i n c e CALC i s a  intervals  of the  i s thus updated a t i n t e r v a l s of  held constant otherwise.  computations on  that  t h e lower  implementation of the c o n t r o l  , the  bound  system  with  a real manipulator. The types  of  using loop,  many  subparts  control  o f d i s c r e t e b l o c k CALC  t o be p e r f o r m e d .  allow open  loop,  only the desired Cartesian path s p e c i f i c a t i o n  or  closed  using  f o r error  t h e view  C o n t r o l c a n be  various  of the r e s u l t i n g  50  j o i n t path  correcting control  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 w h e t h e r  i s open o r c l o s e d l o o p .  Since  Cartesian  coordinates,  feedback  Cartesian  coordinates.  An  resulting  path  by  a  using  path  equivalent  of  Selection  i s made  of  approximate C a r t e s i a n view  vision  the v i s i o n  system or  to  logical  Alternatively,  the  exact  obtained  using  the  analytical direct  variable  EXACT  i s true.  control  observations from  at  observations variable  done  is  i  one  true. can  would be.  c  apply  dynamics,  kinematics  equivalent  Machine  1.  The  VISION.  path  can  be  in  a  real  f e e d b a c k c a n n o t make u s e o f c u r r e n t but rather only  previous.  Given  A delay  observations  of t  ^  i n the  i s i m p l e m e n t e d when that  there  to predict  is a what  logical delay  in  the  current  S u c h p r e d i c t i o n i s done whenever  logical  P o s i t i o n and v e l o c i t y  Finally,  prediction  and a c c e l e r a t i o n o r  of  the  the of  of the C a r t e s i a n  Cartesian  analytical  i n v e r s e dynamics,  inverse  o r by u s i n g  i n v e r s e dynamics,  method o f c a l c u l a t i o n  51  state  is  one c a n c a l c u l a t e t h e t o r q u e  by u s i n g t h e a n a l y t i c a l  consisting  and t h e a n a l y t i c a l  learned  2.  Due t o c a l c u l a t i o n d e l a y  to the manipulator  inverse  Machine  a n d a c c e l e r a t i o n d e p e n d i n g on t h e  l o g i c a l v a r i a b l e USEOBS.  path  learned  variable  Cartesian  using e i t h e r the desired v e l o c i t y velocity  a  the  logical  attempt  PRDICT i s t r u e .  observed  to  a  of  i f  of the r e s u l t i n g path  observations,  variable  c  using  in  kinematics  using  time t  DELAY  observations  resulting  of the r e s u l t i n g path  least  by  be  joint  s y s t e m b a s e d on L e a r n i n g  according  is in  i n f o r m a t i o n must a l s o  c a n be d e t e r m i n e d f r o m t h e r e s u l t i n g  model  application,  specification  i s selected  by  a  Learning logical  variable is  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  false.  DISCRETE  BLOCKS  LEARN  I  AND  LEARN  2  D i s c r e t e b l o c k s LEARN 1 a n d LEARN2 c o n t a i n c o d e t h a t L e a r n i n g M a c h i n e s 1 a n d 2, the t  l  system. r  n  using observations of  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 o f t ^  when  2  respectively,  trains  enabled  by l o g i c a l v a r i a b l e s  LRNINV  and  r  n  1  and  LRNVIS,  respectively. DISCRETE  BLOCK  Discrete  SAMPLE  b l o c k SAMPLE c o n t a i n s c o d e t o c o l l e c t  i n f o r m a t i o n about t h e performance of t h e c o n t r o l executed a t i n t e r v a l s of t all path  simulations.  statistical  system.  It is  „ . t _ was s e t t o be 0.01 s e c f o r samp samp  Information compiled  i n c l u d e s r o o t mean s q u a r e  error, XERMS  = (mean{(x.-x,.) }) 2  VXERMS = (meant ( x . - x , . ) } ) 2  1 / 2  (3.77)  2  (3.78)  ^  1 /  0.1  I  AXERMS = ( m e a n { ( x . - x , . ) } ) ^ 2  i  1  (3.79)  2  ai  r o o t mean s q u a r e p a t h e s t i m a t i o n  error,  EXERMS = ( m e a n { ( x . - x  .) })^  EVERMS = ( m e a n { ( x . - x  .) } )  /  EAERMS = ( m e a n { ( x . - x /  2  2  (3.80)  2  (3.81)  1 / / 2  ei  .) }) ^ 2  1  (3.82)  2  ei  r o o t mean s q u a r e l e a r n e d p a t h e s t i m a t i o n LXERMS = ( m e a n { ( x . - x , - ) } ) / /  (3.83)  .) }) 2  1 / 2  i  (3.84)  .) })  1 / 2  LAERMS = ( m e a n { ( x . - x , /  (3.85)  1  lei  LVERMS = ( m e a n { ( x . - x ,  Lei  2  lei  r o o t mean s q u a r e t o r q u e e s t i m a t i o n TERMS = ( m e a n { ( r . - r ) } ) 2  d  error,  2  2  1 / 2  1 /  52  error, (3.86)  and  maximum  applied  TRQMAX Also  = max{abs()}  compiled  resulting motion  torque,  are statistics  path  are  (3.87)  motion  imposed  about  how  often  i s out of bounds, by  a n d how  as bounds  certain  of  the  descriptions  of  certain  far  the  on a l l o w a b l e  control  techniques  considered. More  detailed  simulation of  the  simulation  technique  3.4  are included i n following  PATH  being  on  the  t o d i s c u s s methods  i t i s necessary  are  generated.  the  manipulator;  such  aspects  particular  control  Two  types  linear  of  t o d e s c r i b e how  of paths  paths  path  a r e used  and c i r c u l a r  controlling  the  specifications  to test  control  of  paths.  PATHS  Linear points  paths  given  simply  consists  of f i v e  time  t  there  0  f t  phase  , then of  v  t  max  n  F  .  a m a x  »-  a n c  *  finally  If the i n i t i a l  2  there  +  < x  n 1  ,x^  v m  ax'  occurs  £i2- in2» ] x  i s no p h a s e  a n c  n 2  ^  <  f i i  x  No m o t i o n  a  between x  >  n  a  s  e  2  , / 2  °^  n  e  until a „ . a max m  constant again  points are close  of  together  when,  < W ^ m a x  of constant,  53  T  occurs  acceleration, P  two  ' f i 2 *  a p e r i o d o f no m o t i o n ,  and f i n a l  This  >  motion  of constant  velocity,  i s not reached.  case  of l i n e a r  of motion.  i s a phase  K«fir*in1» ' this  phases  constant  decelaration, duration  consist  by t h e c o o r d i n a t e s <x^  path  In  to  the  studied.  going  manipulator  then  central  s e c t i o n s when  of  SPECIFICATION  Before  LINEAR  are  aspects  non-zero  ( 3  - > 8 e  velocity,  rather there and  i s simply a c c e l e r a t i o n u n t i l  then d e c e l e r a t i o n u n t i l Four s t a n d a r d  plot  of l i n e  F i g u r e 3.4  along the path  stopped.  l i n e a r paths  data d e f i n i n g these  halfway  were u s e d f o r t e s t p u r p o s e s .  four standard paths  i s given  i n t a b l e 3.1. A  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  shows l i n e  PATH  in1  x  1 in joint  x  in2  X  The  i n figure  3.3.  coordinates.  fi1  X  V  f i2  max  a  0  fc  max  line  1  0.45  -1 .45  -0.45  -0.55  0.80  0.80  0.50  line  2  0.45  -0.55  -0.45  -1 .45  0.80  0.80  0.50  line  3  -1 .00  0.45  -1 .00  0.80  0.80  0.50  line  4  -1 .45  0.00  -0.55  0.80  0.80  0.50  linear  paths  Table  3.1  . -0.45 0.00  Data d e f i n i n g standard  CIRCULAR  PATHS  Circular  paths  a r e c e n t e r e d about the o r i g i n of the  space l o c a t e d a t c o o r d i n a t e s  < x  clockwise or counterclockwise. the s p e c i f i e d  c  i»  x C  2  >  =  <  0»~  The i n i t i a l  r a d i u s , p, a n d t h e s p e c i f i e d  , >  «  Motion  position  work  can  be  i s g i v e n by  starting angle,  6,  follows, x  in1  =  x  c1  x in2 = x  +  c 2  P  c  o  s  (  e  (3.89)  )  + psin(0)  (3.90)  54  as  CO • •  s  s  (_> LU  toe lo X DO  1.60  T SEC g  8 •  —*"  o o LU  LU <"§  Csl X  —"•  X 2o —1  •  1  B§  '0.  0.80  2.40  1.60  3.20  4.00  T SEC  s  S  »  ,  s  s CO  •i  Figure  3.3  1  Standard path  4.00  line  1 in Cartesian  55  coordinates  CM • •  CO  UJ  Odor:  0.00  CD"  0.80  4.00  1.60  T SEC  CO"  o  CJ  UJ  UJ  cn CO £g <E—•-  cx >a  CNJ  —H  3 O* •  CO'  •  o  gB  a - -• —*  X H-CN  • CM"  i g u r e 3.4  CO  0.00  0.80  1.60  2.40  3.20  4.00  1.60  2.40  3.20  4.00  T SEC  •  CO"  gs acM-  5 is  •  0.00  0.80  T SEC  Standard path l i n e 1 i n j o i n t c o o r d i n a t e s  56  Motion  proceeds  along  s p e c i f i e d by / 3 .  complete r e v o l u t i o n five  a c i r c u l a r arc f o r the fraction  phases of motion occur as with  v. „ a n d a.„„ r e p l a c e tng tng c  tangential however,  v max  velocities such  acceleration  that never  a n d a„ , max  exceeds  l i n e a r paths,  except  respectively. r  must  This  that  The maximum  J  be  constrained,  sum o f t a n g e n t i a l  a„ . max  a  A l o n g t h e t a n g e n t , t h e same  and a c c e l e r a t i o n s  the vector  of  and  radial  i s accomplished  by  r  J  choosing, v. tng a  = min { v . ( P 7max amav)l/2} max  ( 3 . 9 1 )  =  ( 3 . 9 2 )  m  r n o  <  a  tng where  „ ,  a  v  2  ~  max  V .  n  r  ,  4  /  2 P  )  l  /  2  tng  7 i sa factor that  specifies the portion  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 tangential Two  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 1 i n Cartesian  3 . 6 shows c i r c l e  shows a p l a n  P  A  T  circle  1  circle  2  that  x  the  is  constant  i n table  The  3.2.  data  A plot  c o o r d i n a t e s i s shown i n f i g u r e 3 . 5 . 1 i n joint  c o o r d i n a t e s and f i g u r e 3 . 7  view of c i r c l e 1 .  P  H  a  v e l o c i t y phase o f m o t i o n .  circle  Figure  during  m  c i r c u l a r p a t h s were u s e d f o r t e s t p u r p o s e s .  defining of  of a  0  /  3  7  v ^„ max m  a  m  a  v  max  t u  n  dir  0 . 4 0 0 . 0 0 1 . 0 0 0 . 7 5 0 . 9 0 0 . 9 0 1 . 0 0 cw • 0 . 4 0  3 . 1 4 1 . 0 0 0 . 7 5 0 . 9 0 0 . 9 0 1 . 0 0 ccw  Table 3 . 2 Data d e f i n i n g  standard c i r c u l a r paths  57-  58  CO"  OJ • •  o  UJ COO  cn  -  O UJ  to •  Q  <r-4  ,  .00  SEC CM  in  x t~«M CM*"  F i g u r e 3.6  Standard path c i r c l e  59  1 in joint coordinates  O  I  o • o" CO  •  •  I  CSJ X.  o CO  oin 0.50  F i g u r e 3.7  -0.30  •0.10  DX(1)  0.10  M  View of s t a n d a r d p a t h  ( t i c k s mark i n t e r v a l s of 0.2  sec)  60  circle  3.5  MANIPULATOR CONTROL USING THE ANALYTICAL  INVERSE  DYNAMICS  inverse  dynamics  AND ANALYTICAL INVERSE KINEMATICS Manipulator and  analytical  control using inverse  analytical  Cartesian  it  a  forms  3.5.1  manipulator.  requires  path.  sum o f p o l y n o m i a l s  It  compare  manipulator  representation  of t h e  CONTROL  simulate  ideal  explanation.  inverse  used  to  control scenarios,  open  loop  control  i t is  of  the  The m e a n i n g o f open l o o p c o n t r o l i s c l e a r , h o w e v e r ,  analytical are  which  m o v i n g on t o more r e a l i s t i c to  c a l l e d the  dynamics.  IDEAL OPEN LOOP  instructive  ideal  which are together  i n v e r s e dynamics, warrants i n v e s t i g a t i o n as  a learned  inverse  Before  kinematics,  benchmark a g a i n s t  control using Cartesian  the a n a l y t i c a l  By  we  mean  that  the  and a n a l y t i c a l  inverse  dynamics  t o compute t h e t o r q u e s r e q u i r e d t o  achieve  a  also  kinematics  ideal,  means t h a t t h e t o r q u e i s u p d a t e d s o  that  i ti s effectively calculated continuously.  open  loop  control,  CLOSED=.FALSE. ,  two  simulations  INVARM= .TRUE.  were  frequently  To t e s t  carried  a n d t: ,„ = 0.0001  given  ideal  out  with  sec. Since the  C31C  duration  o f m o t i o n f o r t h e p a t h s was s e v e r a l  small  ,  t  assures  that  torque  seconds,  i s effectively  such  a  calculated  C cL J. C  continuously. the  paths,  resulting line  line  1  3.3 shows t h e p a t h e r r o r t h a t o c c u r r e d f o r and c i r c l e  1,  and f i g u r e  path i n Cartesian coordinates  f o r the  3.8  shows  simulation  the of  1. The e r r o r s t h a t o c c u r a r e m i n i m a l f o r b o t h p a t h s a n d t h e  resulting path,  Table  path 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 t h e d e s i r e d  w h i c h i s shown i n f i g u r e 3.3. The e r r o r s c a n be a t t r i b u t e d 61  s CM  C_>  0.00  8  0.80  1.60  2.40  3.20  4.00  T SEC  8  —  UJ0  cos  </)§  *—  L  C\J •  —  •  0.00  8  0.80  T SEC  1.60  2.40  3.20  4.00  0.80  1.60  2.40  3.20  4.00  S  8  Xo  0.00  Figure  3.8  Ideal  open l o o p  T SEC  c o n t r o l of  62  line  1  to 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  inverse  kinematics  i n the  and a n a l y t i c a l  inverse dynamics,  simulation  of  calculation  of the a n a l y t i c a l  that  the manipulator,  line  1  circle  T a b l e 3.3  d i r e c t kinematics,  control  1  such  s i m u l a t i o n program.  velocity  3.05E-5  1.05E-4  2.10E-5  3.22E-5  7.78E-5  i d e a l open l o o p  is  AXERMS  control  open l o o p c o n t r o l i s n o t  i t serves First,  to  test  two  a  realistic  aspects  of the  i t confirms that path s p e c i f i c a t i o n The  respectively.  desired  acceleration  Secondly, i t confirms that  and  that  the kinematics  d o e s make i t v e r y  kinematics have  simulation.  the inverse  and i n v e r s e dynamics a r e indeed t h e i n v e r s e  d i r e c t dynamics and d i r e c t k i n e m a t i c s  errors  torque  a r e indeed t h e d e r i v a t i v e s of t h e d e s i r e d v e l o c i t y and  kinematics  it  that  2.54E-5  performed c o r r e c t l y .  position,  mean  the  and t h e f a c t  VXERMS  ideal  strategy,  being  albeit  in  XERMS  Path e r r o r using  While  precision  frequently.  PATH  is  finite  torque i s not updated c o n t i n u o u s l y ,  updated very  inaccuracy  used.  W h i l e t h i s does not  and dynamics a r e n e c e s s a r i l y  probable that,  given  of the  that  the  correct, analytical  a n d a n a l y t i c a l d y n a m i c s h a v e been d e r i v e d c o r r e c t l y , n o t been i n t r o d u c e d If  i n the implementation  e i t h e r o f t h e s e a s s e r t i o n s were  63  not  of the true,  i t  w o u l d 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 the d e s i r e d path  path would d u p l i c a t e  s o c l o s e l y when u s i n g open l o o p  In a d d i t i o n ,  control.  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 e x a m i n e  the torques  t h a t must be a p p l i e d t o t h e m a n i p u l a t o r  through  specified  a  function  of  time,  i t as  F i g u r e 3 . 9 shows  the  torque,  that i s required to drive  the  manipulator  path.  through standard path,  Figure 3 . 9  to drive  line  Torque p r o f i l e  1.  for line  64  1  3.5.2  IDEAL CLOSED The  using  position  ideal  correcting Ideal  open  closed  test  carried  out with  PRDICT=.FALSE., The  fact,  any  precision The  control path  ideal  perturbed  from  t=0.2  correcting  reduction  a  action  the desired  feedback.  Such  theory  i t would  seem  of  such  position  and  desired  a  in  which of  one c o u l d  a  3.4  were  =  c  0  -  0  and  0  0  1  are  control.  <  loop  the  x (  ji'  x (  by  32  In  by  the  achieve with  increasing  control  i s  manipulator  i s  >  =  <  U  V~  position  i s shown  velocity errors simply  c  is limited  to the desired  simulation  that  i  t  loop  of c l o s e d  position  i s restored  error  applied  EXACT=.TRUE.,  in table  open  the  simulation.  simulation  sec and  as  of path  of the  2  ideal  control.  simulations  k =64and  a r e shown  error  of the  INVARM=.TRUE.,  i n the  correcting  two  when  using loop  updating  control,  with  occur  measurement  ^ = 16,  errors  occurring  In  constant  loop  by  closed  assumes exact  closed  that  reduced  and continuous  of c a l c u l a t i o n s  by  errors  in ideal  DELAY=.FALSE.,  those  error  c a n be  CLOSED=.TRUE.,  further  illustrated  time  control  r e s u l t i n g path from  tracking  resulting  Cartesian To  reduced  loop  loop  torque.  CONTROL  and v e l o c i t y  feedback  resulting  sec.  LOOP  >  a  t  by  error  in figure  3.10.  critical as  short  k  and  1  1  damping a k  2  time while  2 maintaining done  as  ideal  the condition closed  loop  k  1  =4k . 2  control  65  In p r a c t i c e  cannot  be  this  achieved.  cannot  be  o  (Do -OTP  COB  (SI  =>g o tn  o m  g  g  0.00  0.20  0.00  0.20  0. 40  0.60  0.80  1.00  0.60  0.80  7.00  T SEC  CM  Xo in  ><oin  F i g u r e 3.10  Error  0.40  T SEC  correcting  action  control  66  of  ideal  closed  loop  PATH line  1  circle  T a b l e 3.4  3.5.3  XERMS  VXERMS  AXERMS  2.37E-6  6.34E-6  1.46E-4  1.56E-6  4.97E-6  1.01E-4  1  Path e r r o r using  REALIZABLE  CLOSED  ideal closed  LOOP  loop control  CONTROL - STANDARD  ANALYTICAL  CONTROL In  a  various  practical  implementation of  constraints necessitate  contraints  typically  have  closed  deviation  their  loop  control,  from i d e a l i t y .  roots  in  These  technological  l i m i t a t i o n s on t h e s p e e d a t w h i c h c a l c u l a t i o n s c a n be p e r f o r m e d . CALCULATION  DELAY  A c e r t a i n amount o f t i m e p a s s e s b e t w e e n t h e o b s e r v a t i o n the  manipulator p o s i t i o n ,  and t h e c a l c u l a t i o n of the torque t o  apply t o the manipulator to correct while  tracking  observation  action.  Such  logical  variable  *o  (  t  )  =  5  a  s e  p o s i t i o n and v e l o c i t y e r r o r s  a path s p e c i f i c a t i o n .  between  (  t  i s implemented i n t h e  DELAY "  t  c a l c  is  torque  calculation desirable possible  will  interval. to  a  simulation  (  be  quantized  when  3  '  9  3  )  or p i p e l i n e d processing, in  time  To k e e p c o m p u t e r c o s t  resulting  delay  correcting  )  permit t h i s c a l c u l a t i o n delay  without  thus  true as f o l l o w s ,  U n l e s s one u s e s p a r a l l e l p r o c e s s i n g applied  There i s  o f m a n i p u l a t o r motion and e r r o r  delay  of  by  this  the same  t o a minimum, i t i s t o be a s  i n poor c o n t r o l .  We  large  have  as  chosen  t ^ =O.Ol c a  sec as a reasonable  c  INEXACT  compromise.  VISION  E x a c t measurement o f t h e m a n i p u l a t o r p a t h f o r u s e i n correcting  feedback  realistic  is  assumption,  an  unrealistic  although s t i l l  assumption.  rather  v i s i o n system estimates only  to  be c o n s i d e r e d e q u a l t o t h e e x a c t p o s i t i o n ,  the  more  x* , s o a c c u r a t e l y a s v  x.  V e l o c i t y and  w o u l d t h e n 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 o f of time.  We t h u s model  the  system as f o l l o w s ,  x (t)  = x (t)  x (t)  - [x (t>  - x (t-t  c  a  l  c  )]/t  c  a  l  c  (3.95)  J (t)  = [i (t>  - x (t-t  c  a  l  c  )]/t  c  a  l  c  (3.96)  e  e  e  Note  t  n  e  i n addition  e  r  e  a  r  (3.94)  e  e  that  calc'  = x(t)  y  e  t  position,  observed p o s i t i o n as a f u n c t i o n  vision  A  o p t i m i s t i c , i s that  the  acceleration  error  to the previously  mentioned  a d d i t i o n a l e f f e c t i v e delays of  e  v e l o c i t y observations  and t  c  a  ^  foracceleration  c  delay  *- \ / C B  of ^  2  c  o r  observations.  PREDICTION  It using  i spossible prediction.  acceleration information should used  t o reduce t h e e f f e c t of feedback Since  one  knows t h e d e s i r e d  when t h e o b s e r v a t i o n s  a r e made,  ^  later.  and  one c a n u s e  this  this  e n d , we  velocity  These p r e d i c t i o n s c a n  i n t h e c a l c u l a t i o n of t h e torque t o apply a t To  make u s e o f  position  then  time  and  p  (  t  +  t  calc  )  x  p  (  t  +  t  calc  )  =  *o  (  " *o  t  (  )+  t  )+  *d  ( t ) t  ^d  calc  ( t ) t  calc  68  +  *d  ( t ) t  calc / 2  2  ( 3  be t ca  i  c  velocity  prediction as follows, 5  by  velocity  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 a n d  be a t t i m e t  later.  delay  ' ( 3  9 7 )  '  9 8 )  Note  that  i t i s a l s o p o s s i b l e t o use t h e observed v e l o c i t y  acceleration velocity was  for p r e d i c t i o n purposes,  and a c c e l e r a t i o n .  the desired  Such use of o b s e r v a t i o n s ,  found t o have c e r t a i n d i s a d v a n t a g e s  FEEDBACK  r a t h e r than  and  however,  a n d was n o t a d o p t e d .  GAIN  The  feedback g a i n s ,  k  and k ,  1  c a n n o t be made  2  large  i n practice.  result  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 t e n d e n c y t o cause  instability  given  analytical inverse  A choice  arbitrarily  o f k = 1 6 a n d k = 6 4 was 1  our choice  control;  of  parameters  closed loop c o n t r o l using  kinematics,  analytical  found  2  for the  to  standard analytical  i n v e r s e dynamics and  realistic  constraints. STANDARD  ANALYTICAL  Standard  analytical  CLOSED=.TRUE., DELAY=.TRUE., t  c a  ^ =0.0l  CONTROL  INVARM=.TRUE., PRDICT=.TRUE.,  s e c . To t e s t  c  control  are  analytical  paths.  The r e s u l t i n g  i s very  poorer, open  though,  loop  standard  path  errors  1. P o s i t i o n a n d  but there  i s a  i s significantly  t h a t which appeared p o s s i b l e using closed loop c o n t r o l .  slight  The  ideal  performance  of  a n a l y t i c a l c o n t r o l shown h e r e f o r m s a benchmark f o r t h e  evaluation similar  or ideal  good  i nacceleration. Tracking  than  simulations  3.11 shows t h e r e s u l t i n g p a t h i n  velocity  of steps  2  control,  for the simulation of l i n e  overshooting  VISI0N=.TRUE, 1  Cartesian coordinates tracking  setting  k = 1 6 , k =64 a n d  USEOBS=.FALSE.,  shown i n t a b l e 3.5. F i g u r e  by  EXACT=.FALSE.,  standard  w e r e done o f t h e s i x s t a n d a r d  i s simulated  of  learned  control.  One s h o u l d  results with learned control,  69  hope  but should  to  achieve  not expect t o  exceed  the performance  PATH  Table  of standard  analytical  control.  XERMS  VXERMS  AXERMS  line  1  5.99E-4  2.29E-3  9.54E-2  line  2  7.09E-4  2.44E-3  9.58E-2  line  3  6.75E-4  2.45E-3  9.92E-2  line  4  6.32E-4  2.42E-3  9.87E-2  circle  1  6.53E-4  1.50E-3  4.73E-2  circle  2  6.18E-4  1.48E-3  4.73E-2  3.5  Path  error  using  standard  70  analytical  control  o o CM  eng  T SEC  F i g u r e 3.11  Standard a n a l y t i c a l c o n t r o l of l i n e 1  71  4.00  3.5.4  C H O I C E OF The  chosen the  PARAMETERS  parameters so  as  system  to  be  DEFINING  used  achieve  in standard good  realizable.  parameters  can  be  analytical  control  STANDARD A N A L Y T I C A L analytical  control  The  demonstrated except  with  the  control  were  constraint  that  appropriateness  of  by  using  simulations  that  individual  CONTROL  the  chosen standard  parameters  are  varied. CALCULATION  DELAY  Calculation It  was  poor  delay  chosen  control,  to  be  as  this  computing  hardware.  of  line  1  t  ^ =0.04  c a  sec,  simulations is  permits  from  0.01  position  and  velocity  cause  manipulator,  an  that  the  avoid  resonance  otherwise damp  c  a  i  3.13 0«02  = c  i n the  must  with  that be  the  manipulators  with  path  standard the  i s not at  vibrational have  vibrations.  72  to  of  s  value,  the  t  c  as  n  d  t  c  a  ^  c  control  =  the  a  these  calc ^"^  however, of  e  for that  acceleration  vibration  4  seen  case  cost  simulations  error  be  in  lower  t-calc^*^  i s q u i t e good,  updated  would  sec,  For  resulting  3.14.show  I t can  the  tracking  effect  torque  3.6.  implementation.  without  and  The  poorer.  excessive  real  implementation  sec,  noticeable oscillation could  t  in a  possible  3.12,  in table  progressively  This  as  respectively.  becomes  a  large  cases  i s shown  increased  as  Figures  f o r the  c  i s unavoidable  2  s  e  there  c  ' is  manipulator.  the  links  of  an  modelled  here.  It  i s assumed  a  reasonably  modes be  of  high  actual  rate  manipulator  specially  to  links,  constructed  to  4.00  4.00  4.00  T SEC  F i g u r e 3.12 t  Standard a n a l y t i c a l c o n t r o l  , =0.02 s e c calc  73  of l i n e  1,  except  that  4.00  4.00  T  F i g u r e 3.13 t  "i "0.03  4.00  SEC  Standard a n a l y t i c a l c o n t r o l sec  caic 74  of l i n e  1,  except  that  4.00  4.00  4.00  T SEC  F i g u r e 3.14 t  Standard a n a l y t i c a l c o n t r o l  , =0.04 s e c caic  of l i n e  1,  except  that  INEXACT  VISION  A c c u r a t e measurement feedback  to  measurement necessary  only.  useful  in  improving  t o invest using  i snecessary f o r  control.  translate d i r e c t l y into tracking  equivalent control  be  of manipulator motion  errors.  i n a very accurate v i s i o n  the a n a l y t i c a l direct  Errors  I t i s thus  system,  kinematics,  ori t s i f good  i s t o be a c h i e v e d . I t d o e s seem r e a s o n a b l e , t h o u g h ,  Cartesian  Cartesian  position  velocity  needs  and  to  be  measured  acceleration,  in  that  accurately.  i f required,  c a n 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 t h e p o s i t i o n . F i g u r e 3.15 shows  a simulation  feedback error  of l i n e  1 i n which exact v i s i o n i s used f o r  p u r p o s e s by s e t t i n g EXACT=.TRUE.  i s shown  i n table  3.6.  The  resulting  I t c a n be s e e n t h a t  exact v i s i o n i snot s i g n i f i c a n t l y  better  than w i t h  a v i s i o n s y s t e m where v e l o c i t y a n d a c c e l e r a t i o n  path  control  with  o u r model  of  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 m e a s u r e m e n t s o f p o s i t i o n . PREDICTION  Prediction The b e n e f i t s prediction and  was u s e d .  a r e shown by a s i m u l a t i o n  The r e s u l t i n g p a t h i s shown i slisted  i n table  3.6.  control.  i n w h i c h no  i n f i g u r e 3.16  Lack of  prediction  i n more p r o n o u n c e d 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  a s i g n i f i c a n t increase  using  prediction.  requires thus  been u s e d i n s t a n d a r d a n a l y t i c a l  of p r e d i c t i o n  the path error  results and  has  t h e two  link  a s compared t o manipulator,  only 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 .  adopted  requires  For  i n path error  a  f o r use minimal  i n standard a n a l y t i c a l  amount  of  76  additional  results  prediction  P r e d i c t i o n was control  computation  as i t and  CM  COo  0.00  0.80  0.00  0.80  0.00  0.80  1.60  T SEC  o o  o 5° —'o 1.60  T SEC  o m  X oin  F i g u r e 3.15  4.00  1.60  T SEC  Standard a n a l y t i c a l c o n t r o l of l i n e  EXACT=.TRUE.  77  1  f  except  that  4.00  4.00  T  F i g u r e 3.16  SEC  Standard a n a l y t i c a l c o n t r o l of l i n e  PRDICT=.FALSE.  78  4.00  1,  except  that  significantly  improves  tracking.  In standard a n a l y t i c a l c o n t r o l , using to  the desired  v e l o c i t y and a c c e l e r a t i o n .  3.17 shows a s i m u l a t i o n  resulting better  path  than  error  with  acceleration  of l i n e  1 with  i s shown i n t a b l e  prediction using  specification.  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  desired  v e l o c i t y and a c c e l e r a t i o n ,  prediction,  already  however,  adequate,  become  not  adequate.  otherwise later.  differ  computational prediction  expense  using  observation  of  is  velocity  of  acceleration little  the desired  no and that  somewhat f r o m t h e  f o r - p r e d i c t i o n . We that i s  inadequate  control  acceleration  i s not  when l e a r n i n g ,  with  The  i t m i g h t be more b e n e f i c i a l  a s a way o f m a k i n g  except  Observation  Tracking  a s a way o f i m p r o v i n g c o n t r o l  Also,  required,  purpose.  i s m a r g i n a l such  t o use t h e 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 see  possible  USEOBS=.TRUE.  desired  When c o n t r o l  implemented  for this  3.6.  the  is  I t i s also  use t h e 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  Figure  the  prediction  is  benefit.  as w i l l thus  an  Hence  be  shown  additional we  v e l o c i t y and a c c e l e r a t i o n  adopted for  use  in standard a n a l y t i c a l c o n t r o l . FEEDBACK  GAIN  With closed the  the  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  loop c o n t r o l ,  feedback gains,  i t i s not p o s s i b l e k  1  and k , 2  the  cancelled inverse  manipulator by  increase  t o achieve ever b e t t e r  control.  and v i s i o n system a r e  the a n a l y t i c a l inverse  kinematics.  Thus  the only  dynamics  the c h a r a c t e r i z a t i o n  79  realizable,  to a r b i t r a r i l y  Because of these d e p a r t u r e s from i d e a l i t y , of  in  non-linearities approximately and  analytical  of the  control  F i g u r e 3.17  Standard a n a l y t i c a l c o n t r o l of l i n e  USEOBS=.TRUE.  80  1,  except  that  system  by  equation  a second with  approximately best  constant t r u e . We  control  defining  standard  clearly  feedback system  t h a t k =16 1  oscillatory  i s not o s c i l l a t o r y  t o ^=4 but  and  position  constraint  where and  line  that k  1  The  k =32,  1  2  =4k  2  can  seen t h a t ^=16  use  in standard a n a l y t i c a l  reasonably  analytical  choosing  realizable,  closed  manipulator.  Standard  reasonable  The  Now  2  shows  control  feedback  k =4.  system are  1 where  the  the  control  i s increased  effect  a  gains  to the slower  feedback  respectively.  It  Standard  the  parameters  3.18  from  correction  of d e p a r t u r e  i s shown i n f i g u r e s  s i m u l a t i o n s w i t h v a r i o u s feedback be  only  from  .  1 i s simulated with  k =32,  chosen  the path e r r o r  errors.  is  2  Figure  as the  2 the  (3.31)  shows a s i m u l a t i o n o f l i n e  g a i n s are reduced  v e l o c i t y and  k = 6 4 gave a b o u t  2  t h a t 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 due of  and  k =l024.  1  3.19  in  control.  1 w i t h k = 6 4 and  differential  as  w i t h the other  analytical  Figure  homogeneous  coefficients  found  becoming  increased.  linear,  i n combination  s i m u l a t i o n of l i n e is  order,  and  The  3.20  and  g a i n s of ^=8, path e r r o r s  gains are l i s t e d  3.21 k =128 2  for  these  in table  3.6.  ^ =64 a r e r e a s o n a b l e c h o i c e s  for  2  control. control  has  been  the parameters  necessary  loop  system  control  analytical  benchmark a g a i n s t w h i c h  81  control  established to  for thus  by  implement the  two  serves  a  link as  t o compare l e a r n e d c o n t r o l .  a  4.00  4.00  T SEC  F i g u r e 3.18  Standard a n a l y t i c a l c o n t r o l of l i n e  k,=64 a n d k = ! 0 2 4 o  82  4.00  1,  except  that  CM  Wo  %8  0.00  0.80  1.60  T SEC  4.00  o o  —'o  0.00  4.00  1.60  T SEC  tn  o o  Xo in 4.00  F i g u r e 3.19 k =4 1  and  k  Standard a n a l y t i c a l c o n t r o l of l i n e = 4 2  83  1,  except  that  o o  o o Csl  CM  COo  CJ COo  CM  is  0.00  0.80  0.00  0.80  0.00  0.80  1.60  2.40  3.20  4.00  2.40  3.20  4.00  T SEC  o o  CJ COo  1.60  T SEC  o in  Xo  tn  F i g u r e 3.20 k 8 1 =  and  4.00  1.60  T SEC  Standard a n a l y t i c a l c o n t r o l of l i n e  k =128 2  84  1,  except  that  F i g u r e 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 of l i n e 1,  k =32 and k =32 0  85  except t h a t  (  PATH  VXERMS  AXERMS  NOTES  Standard  line  1  5 .99E-4  2. 2 9 E - 3  9. 5 4 E - 2  line  1  1 .12E-3  4. 5 3 E - 3  1 .2 0 E - 1  line  1  1 .62E-3  9. 7 0 E - 3  2. 0 6 E - 1  line  1  2 •87E-3  3. 7 1 E - 2  6. 4 5 E - 1  line  1  2 .60E-4  1 .6 4 E - 3  9. 0 2 E - 2  Exact  line  1  4 .57E-3  7. 0 6 E - 3  1 .1 2 E - 1  No  line  1  6 .60E-4  2. 0 5 E - 3  9. 6 6 E - 2  Use  line  1  1 .55E-4  6. 5 7 E - 3  4. 3 8 E - 1  k  1  = 64, k  2  = 1024  line  1  2 .44E-3  3. 0 6 E - 3  8. 7 4 E - 2  k  1  = 4,  k  2  = 4  line  1  2 .47E-4  2. 6 4 E - 3  9. 3 5 E - 2  k  1  =8,  k  2  = 128  line  1  1 .36E-3  2. 9 2 E - 3  1 .0 9 E - 1  k  1  = 32, k  2  = 32  Table for  XERMS  3.6  npted  Path  error  using  standard  variations  86  'calc  =  ° -  0  2  s  e  c  ^ a l c  =  ° -  0  3  s  e  c  ^ a l c  =  ° -  0  4  s  e  c  Vision  Prediction Observations  analytical  control,  except  3.6  ADEQUACY  INVERSE  OF  DYNAMICS  A  AND  principal  polynomials  of of  the  learning  is  to  the  adequately  POLYNOMIALS  goal  of  two  work  the  is  to  without  Before  performed,  learn  i n v e r s e dynamics  manipulator  manipulator. be  a  of  inverse  recourse  considering  how  to such  i t is instructive  to  consider even  and  inverse kinematics  can  represented  sum  of  the  a  THE  sum  and  i n v e r s e dynamics by  OF  KINEMATICS this  link  REPRESENTATION  polynomials  for  be  purposes  control.  3.6.1  DERIVATION  INVERSE  closely The  (3.51)  A  the  straightforward that  OF  SUM  OF  POLYNOMIALS  REPRESENTATION  OF  THE  DYNAMICS  Although  it  OF  INVERSE  the  analysis  whether  SUM  r e p r e s e n t a t i o n of  kinematics  of  A  to  i n v e r s e dynamics  are  derive a  polynomials  approximates  analytical  through  parameters.  To  to  To  analytical  derive a  consider a  this  of  inverse dynamics  (3.66).  i s necessary  the  sum  end  we  quite  two  have  inverse  are  sum  of  link  complex,  a  is  representation dynamics.  given  by  equations  polynomials  equivalent  manipulator  chosen  i t  two  link  with  specific  manipulator  as  follows,  The  m  1  =  1 Kg  (3.99)  m  2  =  1 Kg  (3.100)  1  1  =  1 m  (3.101)  1  2  =  1 m  (3.102)  standard g  For  =  9.81  these  gravitational m/sec choices  acceleration  has  been  assumed; (3.103)  2  of  parameters,  87  the  coefficients  in  the  analytical d d  n  1  d  inverse  dynamics  are, neglecting  u n i t s , as  = 3 + 2cos(a )  (3.104)  =  (3.105)  2  2  n  1 + cos(a ) 2  = 0  i  follows,  (3.106)  d  1  2  2  = - sin(a )  (3.107)  d  1  1  2  = - sin(a )  (3.108)  d  l  2  = - sin(a )  (3.109)  d  2  2  1  2  1  =  I9.62sin(a )  + 9.81 s i n ( a + a )  1  }  (3.110)  2  and, d  2 1  =  1 + cos(a )  (3.111)  d  2  2  =  1  (3.112)  d  2  1  d  With  2  1  2  2  = sin(a )  (3.113)  = 0  (3.114)  2  2  d  2  1  2  = 0  (3.115)  d  2  2  1  = 0  (3.116)  d  2  = 9.81sin(a +a ) 1  these T  1  coefficients  = 3d  the a n a l y t i c a l  + 2cos(a )d  1  (3.117)  2  2  + d  1  +  2  inverse  cos(a )d 2  dynamics a r e ,  2  2  T  2  -  sin(a )d  +  I9.62sin(a )  2  1  +  + cos(a )d 2  sin(a )d 2  The  sine  2  -7r  < a . < 7r to 4  using  1  +  d  2  (3.119)  t e r m s c a n be a p p r o x i m a t e d their  «  1 - a  sin(a )  *  a  2  (3.118)  2  2  cos(a ) 2  2  + 9.81sin(a,+a )  2  order  t h  1  + 9.81 s i n ( a + a )  1  1  and cosine  2  2sin(a )d d  1  = d  truncated  -  2  equivalent  and lower 2 2  - a  /2 3 2  + a  the  range  representations,  terms, 4  2  series  over  /24  /6  (3.120) (3.121)  8 8  sin(  )  a i  = a. -  sin(a^+a ) —  ( 3 . 122)  + a  2  3 /6 -  2  2 Q^^  2 ^  ~ l 2  2  a  a  3 ^  ~ 2 a  2  (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 t h e s i n e and c o s i n e and  keeping the r e s u l t i n g  sum o f p o l y n o m i a l s T  <* 2 9 . 4 a  1  1  - 4.9la 3  T  2  2  * 9.8la  1  2  2  2 2  - 1.64a  Demonstration sum  of  2  - 4.9la  °-2 2 a  +  a  -  2  a  4.9la a  2  1  2  2a  1 ( 3 . 124)  + 9.8la  3 1  2  - 4.9la  2 1  a  -  2  4.9la a 1  2 2  2  0^2°-]  2d.j - 0. 5 a  +  2  of t h e accuracy a c h i e v a b l e  polynomials  a  » ^ 1 ~ 2  2 +  2 1  a  dynamics,  2  3 - 1.64a  2  .2 -  2  d  f o r the inverse  + 9.81a  3 1  . . - 2a a^a  - 0.5a  or lower y i e l d s  1  representation  - 1.64a + 2d  t e r m s o f 4*"* o r d e r  terms  representation  d^ + d with  i s done  (3.125)  2  such a 4 ^ in  the  order  following  sections. 3.6.2  PRE-LEARNING OF THE INVERSE DYNAMICS Having  inverse  d e r i v e d a sum o f p o l y n o m i a l s  dynamics,  representation dynamics  used  could  be l e a r n e d We c a l l  representation  in  any  analagous  to  i s learned  c o n t r o l scheme. that  used  target  using  in  the  analytical  t h i s p r e - l e a r n i n g as the off-line,  T h i s was done chapter  convergence r a t e s of t h e v a r i o u s the  of the  i t seemed n a t u r a l t o d e t e r m i n e w h e t h e r  as a guide.  polynomials  representation  2  to  inverse sum  before  using  a  this  being program  investigate  learning algorithms  of  except  the that  f u n c t i o n s u s e d w e r e t h e two f u n c t i o n s t h a t make  the  i n v e r s e dynamics.  2,  Learning  up  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 M e t h o d  Identification,  f o r the cases,  f = T  1  and f = r  The number o f i n p u t v a r i a b l e s was v = 6 a n d t h e s y s t e m o r d e r 89  2  .  was  s = 4. The i n p u t v a r i a b l e s w e r e , z  1  = a  (3.126)  1  z  2  = a  z  3  = o  z  4  = d  z  5  = a  1  Zg = d  2  Training space,  (3.127)  2  ( 3 . 128)  1  (3.129)  2  •  (3.131)  points -1  were  z.  <  T  Torques  function the  2  a n d 20,000 i t e r a t i o n s  were l e a r n e d  3.22  shows  1  iterations.  estimating  the  reduction  and A f ,  performed.  t o r q u e s do n o t i m p r o v e a f t e r a b o u t  and  1  Af  better  accuracy  s, t h e s y s t e m  shown  in  counterparts proceeds, those  obtained  correspondence analytical  equations  that  the  the  cannot  5000 be  were  i n t e r v a l s of  100  inverse  1  and of  iterations. achieved  in  This without  order.  (3.124) and (3.125) a l o n g pre-learned.  by d e r i v a t i o n . the  a  show  Note t h a t the e s t i m a t e s  Clearly,  c o e f f i c i e n t s are converging  as  as  2  The g r a p h s  T a b l e 3.7 a n d 3.8 shows t h e c o e f f i c i e n t s d e r i v e d and  of  ( i n t h i s case the torques T  functions  the  increasing,  the  I t c a n be s e e n t h a t t h e a v e r a g e e r r o r  the target  that  of A f  averaged over  2  h a s d r o p p e d t o a b o u t 0.05 N»m.  implies  were  iterations.  T ) 2  over  a s sums o f p o l y n o m i a l s  o f t h e number o f t r a i n i n g  magnitudes of A f  training  uniformly  z..  input v a r i a b l e s , Figure  randomly generated  < 1,  and T  1  (3.130)  best  4  order  as  their  learning close  n o t e x p e c t an  approximation  dynamics i s not l i k e l y  90  with  to values  One s h o u l d t h  previously  t o be  to  exact of  exactly  the that  0.00  0.40  O^BO -10^  L20  1.60  2.00  ITERATIONS  o_  F i g u r e 3.22  Reduction  of e s t i m a t i o n e r r o r d u r i n g  pre-learning  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 iteration  interval)  91  100  obtained and  by t r u n c a t i n g  cosine  terms,  Identification approximation  does as  estimation  should  however,  and  pre-learned  as  not q u i t e  error  1  z  1  Z  2  z z 2 1  2  z z 2 1 2  Z  Z  Z  4 3 2 Z  Z  4 %  Z  5  Z  5 2  Z  6  2 Z  of  3.7  the  t  n  order  learning  point.  between  the  There derived  PRE-LEARNED COEFFICIENT  29.4  29.3  -4.91  -4.47  9.81  9.67  -4.91  -4.24  -4.91  -4.23  -1 .64  -1 .33  -2.00  -1.81  -1 .00  -0.92  5.00  5.00  -1 .00  -0.91 1 .99  inverse a n d by  -0.51 <0.06  0.00  Coefficients  derivation  correspondence  4  that  training  -0.50  2  Z  others  Table  by t h e c o n s t r a i n t  sine  Learning  t o the best  2.00  z z 6 2 2  converge  the  Furthermore,  DERIVED COEFFICIENT  Z  Z  here.  of  coefficients.  3  Z  representations  at the last  a close  POLYNOMIAL TERM  z  done'  i t is limited  eliminates be,  the series  for a  sum  dynamics f u n c t i o n pre-learning  92  of polynomials f o r torque  representation T  1  obtained  by  POLYNOMIAL TERM  DERIVED COEFFICIENT  9.81  9.69  -1 .64  -1 .36  9.81  9.71  1  z  3 1 .  z  2  Z  2  2 2 1 2 2 1 Z  Z  Z  Z  2  z 3  3  z  2  Z  Z  Z  Z  -4.91  -4.24  -4.91  -4.22  -1 .64  -1.41  2  5 2  Z  of  3.8  the  Z  6  Coefficients inverse  derivation  3.6.3  inverse  2.00  0.99  0.00  <0.05  for a  dynamics  sum  of polynomials  function  DERIVATION  f o r torque  OF  A  SUM  representation r  2  OF  obtained  by  POLYNOMIALS  OF T H E C A R T E S I A N I N V E R S E DYNAMICS  achieve  to  -0.44  1 .00  ATTEMPTED  our  goal  in Cartesian  Cartesian  attempt  2.00  pre-learning  specification the  0.90  a n d by  REPRESENTATION To  1 .00  -0.50  5 2  others  Table  PRE-LEARNED COEFFICIENT  inverse  derive  kinematics  with  the inverse  the  manipulator  a sum  of  coordinates,  dynamics.  as  Thus  of polynomials  o f t h e two l i n k  dynamics  learned  form  given  i t  i s  It  which  inverse was  path  to  learn  instructive  representation  manipulator,  i n (3.19).  with  i t i s necessary  the Cartesian  93  control  of  to the  together  dynamics of  found  that  an  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  inverse dynamics.could of the space The  g i v e n by e q u a t i o n s or  cosines  dynamics, (3.48)  the manipulator  i s capable.of  k i n e m a t i c s o f t h e two (3.39) t h r o u g h  link  (3.44).  of the a n g u l a r p o s i t i o n s ,  themselves,  is  useful  r a t h e r than the  time  moving.  manipulator  are  Since i t i s the s i n e s  r a t h e r than  that 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 t  Cartesian  o n l y be a c h i e v e d o v e r a p o r t i o n a t a  over which  inverse  the  the  angles  i n t o the i n v e r s e  to consider equations  (3.44)  through  inverse kinematics functions for angular  position. -Given t h e l i n k t h e s i n e s and cos(a ) 2  l e n g t h s chosen p r e v i o u s l y ,  c o s i n e s of a n g u l a r p o s i t i o n = x^/2  + x  2 2  2  = x sin(a )/(x  s i n U ^  = (1 - c o s ( a ) )  ]  well  The  1  two  2  2  1  +x  ) - x /2  2  (3.134)  2  (3.135)  1 / 2  1  functions given here,  r e p r e s e n t e d by a sum  within  (3.133)  l / 2  2  cos(o )  first  ( 3 . 1 32)  2  2  The  become,  /2 - 1  s i n ( a ) = (1 - c o s ( a ) ) 2  the formulas f o r  ( 3 . 1 3 2 ) and  (3.133), can  of p o l y n o m i a l s over the  whole  the m a n i p u l a t o r ' s reach, except.very c l o s e to the  second  two  functions,  r e p r e s e n t a b l e by a sum i n the denominator The  of  ( 3 . 1 3 4 ) and  of p o l y n o m i a l s  (3.135), due  space origin.  appear t o not 2  t o the term  (x  +x  1  be 2 )  2  (3.134).  inverse kinematics functions for d , 1  have t h e term  be  s i n ( a ) appearing 2  i n a denominator  d , 2  d  and  1  d a l l  position.  means t h a t t h e s e f u n c t i o n s h a v e s i n g u l a r i t i e s a t t h e o r i g i n  2  This and  a t t h e l i m i t s o f t h e m a n i p u l a t o r ' s r e a c h where s i n ( a ) i s z e r o . 2  The  singularities  a t t h e o r i g i n and 94  a t t h e l i m i t s of  the  manipulator's inherent good  reach  with  control  Cartesian  control  the m a n i p u l a t o r . at  these  coordinates.  manipulator The  reflect  One  points One  difficulties  s h o u l d not expect  using  would  path  in  normally  not  origin  near  and  have in  utilize  with  the  sum  of  Even n e g l e c t i n g p o i n t s near  the  1  i s more s i g n i f i c a n t .  to  space.  representing cos(a )  polynomials  are  specification  at these problematic p o i n t s i n the  difficulty  that  the l i m i t s of t h e m a n i p u l a t o r ' s  a  reach,  i t does  n o t a p p e a r 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  manipulator's  It  polynomials  over  seems  t h a t any  i s bad  i n the opposite quadrant.  a  circular  band  manipulator's limit  sum  the r e s t of the  of p o l y n o m i a l s t h a t i s g o o d i n one Attempts  e x c l u d i n g the o r i g i n  reach  proved  l e a r n i n g of the  futile.  L I M I T A T I O N OF  CARTESIAN  INVERSE  OF  over  1  and  the  i n v e r s e k i n e m a t i c s and  A SUM  quadrant  to learn cos(a )  I t was  i n v e r s e dynamics t o a p o r t i o n of t h e 3.6.4  reach.  limits  thus  of  the  necessary  to  hence the C a r t e s i a n  space.  POLYNOMIALS REPRESENTATION OF  DYNAMICS TO  A PORTION OF  THE  THE  MANIPULATOR'S  SPACE To  permit  Cartesian to  a  sum  of  i n v e r s e dynamics,  polynomials i t was  a p o r t i o n of the m a n i p u l a t o r ' s  restriction revolute complete  of  joints circle  a restricted  the  representation  necessary  space.  T h i s i s not  t e c h n i q u e a s most r e a l  of motion.  One  origin.  the  learning a  manipulators  that are c o n s t r a i n e d to w i t h i n a f r a c t i o n  great have of  a  n o r m a l l y uses the manipulator  in  r e g i o n or workspace, t y p i c a l l y  the manipulator  to l i m i t  of  Further,  95  one  can  i n f r o n t of or use  several  below  different  sum  of  polynomials  dynamics,  each  representations of  for ' a  different  the  Cartesian  p o r t i o n of  the  inverse  manipulator's  space. We of  restricted  t h e two  link  our  manipulator,  -0.5  < x  1  <  -1.5  < x  2  < -0.5  Velocity  and  attention  0.5  to a workspace  namely  the  below  region  the  origin  specified  m  by,  (3.136) m  (3.137)  acceleration  were  restricted  to magnitudes  of  less  2 than  1 m/sec  variables  Over  and  1 m/sec  f o r the  ,  respectively.  learning  algorithms  The  were  normalized  input  thus,  z  1  =  2x  z  2  =  2x  z  3  = x  z  4  = x  2  (3.141 )  z  5  = x  1  (3.142)  z  6  = x  2  (3.143)  this  represent The  (3.138)  1  +2  2  (3.139) .  1  p o r t i o n of  (3.140)  the manipulator's  the C a r t e s i a n cosine  polynomials  of  a  2  reach  inverse dynamics  i s already  as  i t i s possible  sums  of  exactly represented  to  polynomials. as a  sum  of  of C a r t e s i a n c o o r d i n a t e s ,  cos(a )  = z^/6  The  sine  a  hand  s i d e of  2  of  2  + z  2 2  i s equal  /6  - z /2 2  -  1/2  to the binomial  2  expansion  of the  right  (3.133), 7  sin(a )  (3.144)  =  1 -  cos(a ) 2  4 /2  cos(a )  -  2  6 /8  - cos(a ) 2  /16  ...  (3.145)  A the  good  approximation  series  (3.145)  for  a t an  sin(a ) 2  can  appropriate 96  be  obtained  length,  by  truncating  substituting  i n the  expression  forcos(a ),  and then  2  retaining a l l polynomial  o f a g i v e n maximum o r d e r o r l e s s , difficult  to  do,  however,  (3.145) have s i g n i f i c a n t  terms  say 4 ^ order. This i s rather fc  a s a l a r g e number o f t h e t e r m s  of  c o m p o n e n t s o f 4*"* o r d e r o r l o w e r . 1  Over t h e workspace t h e c o s i n e of a cos(a.,) = 1/2 - z / 4 + z 2  is,  1  sin(a )/(z  1  2  2 1  /2+z  2 2  /2-2z +2) 2  (3.146) The  troublesome  replaced origin  polynomial  with  i n t h e denominator o f (3.146) can  i t s Taylor series equivalent  about  the  of t h e workspace,  (z  2 l  /2+z  2 2  /2-2z +2)"  = \/2  1  2  + z /2 - z^/8 + 3z 2  - 3Z The  taken  be  expression f o rcos(a^)  or lower. Given  of c o s ( a ) ,  one c a n t h e n  1  expansion  Z /12 + 3Z  can then  of 4 k o r d e r fc  2 ]  2  3 2  /12  2 2  /8  ... ( 3 . 1 4 7 )  be t r u n c a t e d t o t h o s e  terms  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  obtain the sine of  using a binomial  o f t h e r i g h t hand s i d e o f (3.135) ,  a s was done f o r  s i n (a ), 1  sin(a ) = 1 - cos(a ) /2 2  2  2  -  cos(a ) /8 4  2  - c o s ( a ) / l 6 ...  (3.148)  6  2  The  factor  functions series  2  ford , 1  d , d 2  1  that appears i n the i n v e r s e kinematics and d  f o r the reciprocal  sin(a ), 2  taken  Accurate terms  l/sin(a )  can  truncated  c a n be r e p l a c e d w i t h t h e T a y l o r  2  of t h e polynomial approximation  about t h e o r i g i n  of the workspace.  r e p r e s e n t a t i o n of these  sine,  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 t o a c e r t a i n maximum o r d e r ,  Substituting  these  results  for  into  97  c o s i n e and cosecant s e r i e s shown  4*"** o r d e r the  inverse  here,  f o r example. kinematics  relationships  and subsequently  with  dynamics,  inverse  polynomials Again,  expression  only  retained.  It  necessary the  terms i s  of  clearly such  inverse  representation sum  for  the  an  maximum  difficult  be  algebra  representation  however, of  need  the  The f e a s i b i l i t y  representation  of  dynamics.  or less  perform  a sum o f p o l y n o m i a l s dynamics.  sum  inverse  order  to  kinematics  accurate  Cartesian  c a n be d e m o n s t r a t e d ,  polynomials  the inverse  can obtain  of a s c e r t a i n  to derive  Cartesian  one  combining  of  of  such  a  by p r e - l e a r n i n g  the  Cartesian  a  inverse  dynamics. 3.6.5  PRE-LEARNING To  OF T H E C A R T E S I A N  demonstrate  representation  of  representation  was  inverse  the  dynamics  analagous  to  convergence the  target  the  Cartesian  adequacy  Cartesian  pre-learned  used  in  inverse  used  sum  of  polynomials  dynamics,  such  the a n a l y t i c a l  was d o n e  chapter  were  DYNAMICS  inverse  This  of the various  functions  of  using  as a guide.  that  rates  the  INVERSE  2  learning  a  program  investigate  algorithms  t h e two f u n c t i o n s  dynamics.  Cartesian  using  to  Pre-learning  the  except  that  was  a  that  make  up  carried  out  * using  Method  2,  Learning  Identification,  f o r the cases  f = r  1  * and  ~  f  system  2'  T  order  equations  T  n  e  was  number s = 4.  (3.138)  uniformly  iterations  were  function  of  input  The i n p u t  through  generated  Figure  °f  (3.143).  over  variables variables Training  the space,  -1  was v = 6 were  and  those  points  given  were  < z. <  1,  of A f  and  the in  randomly  and  20,000  performed.  3.23  shows  t h e number  the  reduction  of t r a i n i n g 98  iterations  1  A f  2  performed.  as  a  The  0.40  0.00  the  JL20  «io« ITERATIONS  F i g u r e 3.23 of  0JBC)  Reduction  of e s t i m a t i o n e r r o r d u r i n g  C a r t e s i a n i n v e r s e dynamics ( e s t i m a t i o n  over each 100 i t e r a t i o n  2.00  1.60  interval)  99  pre-learning  error  averaged  graphs  show  the  magnitudes  intervals  of  estimating  the target  r )  has  2  Cartesian  CLOSED  INVERSE  about  control  LOOP  CONTROL  test  the adequacy  Cartesian  inverse  inverse  was  to that  error T  the torques found  here,  in 1  that  were  using  USING  THE  the  and the  adequate  analytical  PRE-LEARNED  were  CARTESIAN  coefficients torques.  resulting tracking  k =64  used  in  carried errors  control.  3 mm  i  =  0  c  «  u  1  s  in estimating each  Simulation  n  pre-learned  e  1  to  the  3.9.  Also  The e r r o r  of path  simulations. paths.  mm  when  accelaration in table  maximum  i n the  3.9  torque  estimating  o f t h e maximum 1.  The  position  0.5  shown  and the  estimate  Mean  t o about  4 percent  100  T  V e l o c i t y and  the torque  shows a p l o t  •  standard  in table  increased.  about  c  during  compared  path.  e  Machine  place  control.  error  3.24  a  out f o r a l l s i x  analytical  i s typically  c  a r e shown  i s about  during  t  Learning  took  is  Figure  analytical  and  2  errors are similarly  torque  analytical  PRDICT=.TRUE.,  were  i s applied  substituted f o r the  DELAY=.TRUE.,  tracking the  pre-learned  VISION=.TRUE.,  learning  standard  the  INVARM=.FALSE.,  were  error  were  out i n which  inverse  CLOSED=.TRUE.,  1  path  carried  standard  k =16,  No  Simulations  were  Cartesian  s e t as f o l l o w s :  EXACT=.FALSE., USEOBS=.FALSE.,  of the pre-learned  dynamics  i n otherwise  parameters  torque.  It  over  dynamics.  simulations  the  case  as pre-learned  comparable  dynamics,  that  N.m.  averaged  2  The average  (inthis  0.6  Af ,  DYNAMICS  To  using  and  1  iterations.  dynamics,  inverse  Af  functions  to  inverse  achieve  3.6.6  training  dropped  Cartesian to  100  of  applied  I t i s evident  that  F i g u r e 3.24 Cartesian  Closed  l o o p c o n t r o l of l i n e  i n v e r s e dynamics  101  1 u s i n g the  pre-learned  control than  i s quite  some  is  i t  polynomials adequate  good  large glitches  of steps  in  acceleration.  as good as w i t h  enough  to demonstrate  good  other  While  standard  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  to achieve  are occuring  that  a  sum  inverse dynamics  XERMS  VXERMS  AXERMS  TERMS  TRQMAX  3. 8 2 E - 3  7. 2 6 E - 3  9. 9 4 E - 2  3 • 44E-1  8.33  line  2  8. 8 8 E - 3  2. 0 2 E - 3  1 .36E-1  4 .08E- 1  9.15  line  3  6. 0 5 E - 4  2. 7 6 E - 3  1 .01E-1  1 .65E- 1  14.1  line  4  3. 3 2 E - 3  6. 4 9 E - 3  1 .01E-1  4 . 5 6 E -1  6.79  circle  1  2. 4 6 E - 3  6. 1 9 E - 3  5. 3 0 E - 2  3 .21E- 1  9.62  circle  2  2. 0 1 E - 3  6. 4 1 E - 3  5. 5 6 E - 2  2 .68E- 1  13.6  3.9  Path  and  torque  error  inverse dynamics i n c l o s e d  ADEQUACY  DIRECT  error  OF  POSITION  Another polynomials  A  kinematics,  is  OF  using  loop  POLYNOMIALS  the  pre-learned  control  REPRESENTATION  OF  THE  KINEMATICS  principal  goal  of t h i s  work  r e p r e s e n t a t i o n of the d i r e c t  manipulator  manipulator.  SUM  of  control.  1  Cartesian  the  analytical  line  Table  link  no  not q u i t e  is  PATH  3.7  as  overshooting  performance control,  stable  As i t  without with  is  the  recourse inverse  instructive  kinematics to  a  consider  sum  of the  analysis  dynamics  to first  102  i s to learn  and  of  of two the  inverse  whether  the  direct sum  position  of  k i n e m a t i c s c a n be a d e q u a t e l y  polynomials  f o r t h e purpose of  represented  replacing  the  by  a  vision  system. 3.7.1  DERIVATION OF A SUM OF POLYNOMIALS  REPRESENTATION  OF  THE  DIRECT POSITION KINEMATICS A  sum o f p o l y n o m i a l s  kinematics  can  be  r e p r e s e n t a t i o n of the d i r e c t  derived  quite  k i n e m a t i c s o f any m a n i p u l a t o r by  a  general  polynomials achievable The  analytical  representation  as  direct  =  x  2  = -cosla^)  1  are  - cos(a  z  2  <z ,z > 1  the With  2  sum  of  kinematics  is  kinematics are  given  manipulator  (3.99) t h r o u g h  by with  (3.103),  the  follows,  + a )  (3.149)  1  + a )  (3.150)  2  2  modelling  the  the workspace i n p a r t i c u l a r ,  direct  position  the input  variables  r e p r e s e n t a t i o n c a n be c h o s e n a s  follows,  = (a, + ir/3)/U/2)  (3.151)  - 27r/3)/(7r/2)  (3.152)  = (a  2  = <0,0> c o r r e s p o n d s  workspace i s e n c l o s e d such  position  a  form  1  concerned with  sum o f p o l y n o m i a l s }  that  direct  F o r a two l i n k  i n equations  sin(a ) + sin(a  we  direct  in a similar  appear  the  position  kinematics are as  1  z  direct  given  k i n e m a t i c s over a  of  the  for a l l manipulators.  x  Since  i t would  (3.32) and ( 3 . 3 3 ) .  parameters  Since  c a n be o b t a i n e d  technique,  analytical  equations  easily.  position  t o t h e center of t h e workspace i n t h e r e g i o n d e f i n e d by -1 < z.  a n o r m a l i z a t i o n of the input  kinematics  become,  103  variables  the  and < 1.  direct  x  = [ s i n ( 7 r z ^/2) - i / 3 c o s ( i r z ^/2) +  1  + c o s (7T2 ^2)5111(7:22/2)  5^(7^/2)005(7^2/2)  + ^3005(7:2^2)005(7:22/2)  - l / S s i n d r z / 2 ) s i n ( i r z / 2 ) ]/2  (3.153)  2  x  2  -  = [ ~ c o s ( 7 : z / 2 ) - i/3sin(7T2^/2) 1  + s i n ( T : Z / 2 ) s i n (irz /2)  +  2  1  005(7:2^2)005(7:22/2)  v 3sin(7:z /2)cos(7:z2/2) /  1  + / 3 c o s ( 7 ^ / 2 ) s i n (7:z /2 ) ]/2  (3.154)  2  The  sine  a n d c o s i n e t e r m s c a n be r e p l a c e d w i t h  equivalents, truncated t o 4 ^ order  o r lower  fc  s i n ( j r z / 2 ) « *z /2  Z  <* 1 - T : 2 Z  cos(-irz /2) 1  -  s i n ( f f 2 / 2 ) * KZy/2 2  2  2  Substituting  sum  of  these  (3.155)  3  /4  + ir z 4  T:3Z  3 1  « 1 - % z /k  cos(irz /2)  (3.154),  2 1  those  polynomials  /384  ( 3 . 1 56)  /48  ( 3 . 1 57)  expressions  and keeping  4 1  + 7^2^/384  2  series  terms,  - v z, /4B  y  1  their  into  terms o f 4  representation  (3.158) equations  t h  of  (3.153) and  order o r lower  yields a  the direct  position  kinematics, x  = i r Z j / 2 + T:2 /4 - V 3 T : 2 2 / 8 - 3 T : 2 / 1 6 /  1  2  /  2  -  1  T: 2 /96 - T : Z 3  3  3  2  + I/3T:4Z1Z /192 3  2  1  2  2  2  >  +  2  y  2  + V/3T: 4 Z 1 2 Z  2 2  3  2  - Tp z z /Z2  Z /32  n z^ /48  2  v  / v  3  3T:4Z 2 /192 3  2  / 1 28 + I / 3 T : 4 Z / 7 6 8 4  2  (3.159) x  2  = -1 + ]/3nz /A 2  + T : 2 / 8 + % z /\S> 2  2  2  - y/ZtC'z  2  1  -  V 3T: 2 2 /64 + T: 2 2 /8 - V 3T: 2 /96  -  T: 2 2 /192 - T : 4 Z 1 2 Z  /  3  2  1  4  2  2  /  1  2  3  1  2  3  2  2 2  3  /128  - T:4Z1Z  3 2  z /Z2  2  2  - 7^2^/384 /192  -  T:4Z /768 4  2  ( 3 . 160) To  test  kinematics, analytical  the accuracy simulations  control  of were  the derived carried  out using  f o r the s i x standard paths  104  direct  position standard  while using the  derived  direct  successive velocity  position  points and  in  path  Estimates  using  estimates  view  a r e shown  position  the  the  modelled  vision  Errors  i n path  3.11.  path  1  Table  compare in  estimation  Figure  circle  obtained  various  system  vision  the  system,  by  simple  3.10 s h o w s t h e  standard well  paths.  with  which by  3.25 s h o w s  using  the the  those  position vision  resulting  derived  LXERMS  LVERMS  LAERMS  line  1  4.89E-3  7.75E-3  5.25E-2  line  2  3.43E-2  3.64E-2  1.09E-1  line  3  8.69E-4  3.42E-3  4.82E-2  line  4  1 .28E-.2  1.65E-2  6.92E-2  circle  1  6.60E-3  1.53E-3  6.08E-2  circle  2  6.58E-3  1.43E-3  5.70E-2  3.10  position  at  direct  kinematics.  P A T H  Table  are  estimates.  for  the position  the modelled  and a c c e l e r a t i o n  in table  standard  to estimate  estimates  estimation  are exact.  of  As w i t h  of the p o s i t i o n  of v e l o c i t y  obtained  system  i n time.  acceleration  differentiating error  kinematics  Path  estimation  error  kinematics  105  using  the  derived  direct  o in  o o' • - .  •o" CM  I  X UJO  o to  o in •0.30  ).50  Figure  3.25  View  •0.10  LEX(l)  of c i r c l e  0,10  M  0.30  0.50  1 using the derived d i r e c t  kinematics  106  position  PATH  Table has  0.0  3.01E-3  4.62E-2  line  2  0.0  3.02E-3  4.64E-2  line  3  0.0  3.21E-3  4.80E-2  line  4  0.0  3.20E-3  4.56E-2  circle  1  0.0  2.90E-3  2.56E-2  circle  2  0.0  2.89E-3  2.57E-2  exact  Path  estimation  position  Having direct  could  to  convergence  that  rates  target  of input  4. T h e i n p u t (3.152). space,  used  in  vision  system  which  KINEMATICS  used  was u s e d  were  representation  was  verified  using  This  2  the analytical  direct  to  using  Method  points  < z. <  those  were  given  randomly  1, a n d 3 0 , 0 0 0  f = x  107  that  1  the that  make  up  Learning  a n d f = x « The 2  order  i n equations generated  iterations  program  except  2,  v a r i a b l e s was v = 6 a n d t h e s y s t e m  v a r i a b l e s were  a  investigate  t h e two f u n c t i o n s  f o r the cases,  the this  learning algorithms  kinematics.  of  that  was d o n e  chapter  of the various  Training -1  i t  as a guide.  position  Identification,  the  POSITION  be p r e - l e a r n e d  functions  direct  using  a sum o f p o l y n o m i a l s  kinematics,  kinematics  analagous  the  OF T H E D I R E C T  position  position  error  estimation  derived  representation  number  LAERMS  1  3.11  the  LVERMS  line  3.7.2 P R E - L E A R N I N G  the  LXERMS  were  was  s =  (3.151) and  uniformly  over  performed.  Figure Af  1  and  graphs  Af , 2  show  intervals  x > 2  shows t h e r e d u c t i o n o f t h e e s t i m a t i o n  a s a f u n c t i o n o f t h e number o f the  of  estimating and  3.26  magnitudes of  100  training  with  i t e r a t i o n s were n o t r e d u c i n g T a b l e 3.12 l i s t s derived coefficients  Af ,  The  averaged  2  The a v e r a g e  over  error  This  a  order  represents system  the  as  would direct  position  occurred Table  from e q u a t i o n s  further  (3.159) and (3.160).  I t can  in  coefficients,  estimation e r r o r s using the  kinematics  would  those  be s i m i l a r  to  one  pre-learned those  that  p r e v i o u s l y with the derived d i r e c t p o s i t i o n kinematics.  p a t h s when s i m u l a t e d  viewed  Figure  1  c o e f f i c i e n t s alongside the  Given the s i m i l a r i t y  3.13 shows t h e e r r o r s i n p a t h  standard  x  the e r r o r .  the pre-learned  expect that path  in  best  seen t h a t t h e l e a r n e d c o e f f i c i e n t s a r e q u i t e c l o s e t o  derived previously.  and  iterations.  ( i n t h i s case the p o s i t i o n s  d e c r e a s e s t o a b o u t 0.03 m. achievable  and  1  iterations.  the target functions  accuracy  be  Af  errors,  with  using  the pre-learned  estimation standard direct  the  analytical  position  3.27 shows t h e r e s u l t i n g v i e w o f c i r c l e  108  for  1.  various control  kinematics.  ex  -  o -  F i g u r e 3.26 of  the  Reduction  of e s t i m a t i o n e r r o r d u r i n g  d i r e c t p o s i t i o n kinematics  over each 100 i t e r a t i o n  interval)  109  (estimation  pre-learning  error  averaged  110  POLYNOMIAL TERM  1 z  1 2  1 Z  2  /  - I Z  2  Z  2 1  Z  2 1  Z  2  z  2  2  2 Z  z 2  1 2  Z  2  3  z 1 Z  h  z  4  -1 .02  0.020  -1 .00  1 .57  1 .50  0.0  0.0  0.044  1 .23  1.18  0.0  0. 1 33  -0.496  -0.082  0.0  0.056  -0.254  -0.226  0.785  0.730  1 .36  1 .26  1 .23  1.14  -1 .68  -1 .12  -0.507  -0.398  0.617  0.580  -2.14  -2.04  -0.969  -0.668  0.879  0.613  -1 .07  -1 .02  -0.969  -0.667  -0.839  -1.16  0.984  -0.761  -0.542  -0.323  -0.221  -0.559  -0.364  0.879  0.632  -0.507  -0.380  0.220  0.146  -0.127  -0.112  1 .32  z  X(2) COEFFICIENTS DERIVED PRE-LEARNED  0.0  -0.646  4  z  X(1) COEFFICIENTS DERIVED PRE-LEARNED  2  Table  3.12  Coefficients  polynomials functions  for  representations  f o r p o s i t i o n s x. a n d  derived  and p r e - l e a r n e d  of the d i r e c t x  9  111  position  sum  of  kinematics  PATH  Table  LAEPvMS  1  3.17E-2  1.59E-2  5.78E-2  line  2  4.49E-2  3.11E-2  7.67E-2  line  3  3.74E-2  2.52E-2  7.00E-2  line  4  3.27E-2  1.79E-2  5.37E-2  circle  1  3.74E-2  2.18E-2  4.85E-2  circle  2  3.73E-2  2.40E-2  4.98E-2  position  3.7.3  Path  INVERSE  estimation  LOOP  DYNAMICS  Simulations  AND  position  follows:  CONTROL  USING  PRE-LEARNED  verified  the pre-learned  direct  error  using  the pre-learned  direct  kinematics  CLOSED  that  Cartesian  kinematics.  k~=64  simulated.  The  , =0.01  resulting  path  DIRECT closed  PRE-LEARNED POSITION  loop  inverse  dynamics  sec.  i s  possible  pre-learned were  set  as  EXACT=.FALSE., USEOBS=.FALSE.,  A l ls i x standard  e r r o r s a r e shown  112  and  parameters  PRDICT=.TRUE.,  CARTESIAN  KINEMATICS  control  INVARM=.FALSE.,  DELAY=.TRUE., and t  THE  Simulation  CLOSED=.TRUE.,  VI SION=.FALSE., k=16,  LVERMS  line  3.13  using  LXERMS  paths  in table  were  3.14.  PATH  Table  VXERMS  AXERMS  line  1  2.77E-2  2.27E-2  1.78E-1  line  2  4.08E-2  6.13E-2  3.10E-1  line  3  3.66E-2  5.87E-2  3.16E-1  line  4  2.82E-2  2.83E-2  1.87E-1  circle  1  3.52E-2  3.24E-2  1.56E-1  circle  2  3.37E-2  3.19E-2  1.41E-1  3.14  dynamics loop  XERMS  Path  error  using  and pre-learned  the pre-learned  direct  position  Cartesian  kinematics  inverse  in  closed  control  Tracking vision  system,  inaccuracies position  control  as  follow the path the resulting  followed  closely.  different, through trajectory  achieved  direct  position  order  i t i s forcing  path  as  table  the circle  i n figure  the 3.29.  trajectory  follows the path  actual  specification.  113  Path  position The  the manipulator  to  figure  pre-learned 1  path  that  to  be  pre-learned  i s  the  3.28  direct  appears  The m a n i p u l a t o r such  the  3.12.  F o r example,  using  to  kinematics.  of magnitude  previously in  with the  due  to the inaccuracies i n the  kinematics,  distorted  that  i s t o be e x p e c t e d  a s seen  Standard  Due  a s shown a  that  than  specification.  path  kinematics.  position  shown  perceives  position  direct  this  a r e o f t h e same  errors,  system  closely  however,  worse  of the pre-learned  errors  estimation  shows  i s significantly  somewhat must  move  perceived  r  0.50  Figure  -0.30  3.28  position the  View  0.10  LEX(l) M  of  circle  1 using  0.30  the  position  Cartesian  inverse  kinematics  114  0.50  pre-learned  kinematics during closed loop control  pre-learned  direct  -0.10  dynamics  of c i r c l e and  direct 1 using  pre-learned  o  in  tn ^O.SO  -0.30  -0.10  0.10  XU)  F i g u r e 3.29 circle  View  of  circle  0.30  M  1 during closed loop  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  pre-learned direct p o s i t i o n  0.50  kinematics  115  control dynamics  of and  Figure line  1.  the  sudden  3.30  The  switches  be  large  from  direct  eliminated  the d i r e c t  pre-learned This Since  the to a is  by  using  position direct  perceived  the pre-learned simulation improved  where  the pre-learned  vision  and  then  system  In  because  manipulator dynamics  and  position  kinematics.  short  the direct  avoids  switching 3.31  how  shows  tracking  affected  away  a t t h e end of a  error  a r e used  with  can  the  c o r r e c t i v e path  path be  vision  using  the  purposes. and  3.31  Cartesian  only  true.  this  Notice  kinematics  position  purposes.  t o be  Figure  there  are  several  i n the acceleration p r o f i l e s .  learned  i s doing  a  Learning  Identification  but minimally  position  glitch  learning  subsequently  technique.  one  t o use of the  point,  kinematics.  point  the f i n a l  f o r feedback  glitches  the  direct  3.30  when  Similar glitches  executing  figures  unexplained  this  or  feedback  Learning  to  accurate  undesirable  v a r i a b l e ADJVIS  error  position  point.  by m e a s u r i n g  system  position  the t r a i n i n g  the t r a i n i n g  for  and  when  less  switching  of  a r e due  of a d d i t i o n a l  a t the t r a i n i n g  1 using  from  avoided  error  direct  of l i n e near  kinematics  Minimization  to the  Minimization  before  logical  occur  results  This  iteration  kinematics  setting  estimation  sudden  system  Interference  position  by  that  kinematics.  one  f o r the simulation  initially  error  vision  position  Interference  eliminate  position  to perform  i s invoked  path  accelerations that  the accurate  Identification of  the r e s u l t i n g  perceived  pre-learned can  shows  inverse  a nominal kinematics  I t appears  116  that  These  that  on  the  the learned  at certain  occur  drives  job of c a n c e l l i n g based  other  points  the direct direct  in  path  space,  the  causing  feedback  errors  until  Clearly, less of  mismatch  i f the  accurate, path  space  Nevertheless, possible  using  pre-learned require direct  to°become  such  the the  position  an  path  and  positive,  pre-learned position  show  that  Cartesian  kinematics.  improvement  could  in the  kinematics.  117  space  are  large  not  be  inverse  Finer  accuracy  left.  were  over  coarse  is  reinforcing  kinematics  unstable  specifications  components  briefly  in path  position  become  simulations  direct  thus  points  direct  c o n t r o l would and  inverse  problematic  pre-learned  direct  only  i n the  much  regions  followed. control  dynamics  c o n t r o l appears of  the  is and to  learned  o o CM • • U J  —1  -1  0  Sg 0.00  F i g u r e 3.30 Cartesian  Closed inverse  0.80  1.60  T  loop control dynamics  2.40  SEC  of l i n e  and  kinematics  118  3.20  4.00  1 u s i n g the  pre-learned  pre-learned  direct  position  oo  CM O  COo ,  0.00  s  0.80  1.60  2.40  3.20  4.00  1.60  2.40  3.20  4.00  1.60  2.40  T SEC  s  c_> I±J cos  0  0.00  0.80  0.00  0.80  T SEC  s in  F i g u r e 3.31 Cartesian  4.00  T SEC  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 t h e p r e - l e a r n e d inverse  dyanmics  and  k i n e m a t i c s w i t h ADJVIS=.TRUE.  119  pre-learned  direct  position  3.8  SELF-LEARNING Having  demonstrated  representation achieve  good  learning analysis  i t  Cartesian  manipulator A  motion;  METHOD  hence  FOR  We  DYNAMICS  sum  inverse  inverse  of  a  remains  of the manipulator. observation  INVERSE  that  of the C a r t e s i a n control,  the  through  3.8.1  OF T H E C A R T E S I A N  dynamics  only  applied  polynomials i s adequate  to find  dynamics  have  of  a  method  without  found  that  torques  of  recourse  to  c a n be  done  this  and  to  corresponding  t h e name s e l f - l e a r n i n g .  SELF-LEARNING  OF  THE  CARTESIAN  INVERSE  DYNAMICS THE  METHOD  Given  sufficient  position, use  and v a r i e d  v e l o c i t y , torques  Interference  or r e l a t e d methods  Cartesian  inverse  velocity,  a c c e l e r a t i o n and torque  and  direct  system,  kinematics,  i s  dynamics;  The c o r r e s p o n d e n c e  n-tuple  relationships  i s  imposed  i e . the  t h e same a s t h a t  an  of  also  of the manipulator  a n d 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  Minimization dynamics.  observations  a  valid  dynamics  the  Cartesian  valid  n-tuple  and  for  for  the  position,  by t h e d i r e c t  by t h e  observations  learn  between  manipulator  imposed  to  vision inverse  the  the  direct inverse  relationships. To such  learn  as that  conditions corresponding Here and  t h e z.  the Cartesian  defined must to  inverse  by e q u a t i o n s  be met:  First,  a l l regions  are the normalized  accelerations  that  dynamics  (3.138)  over  through  one must  a path (3.143),  obtain  of the path  space,  Cartesian  positions,  a c t as input  120  variables  in  space two  observations -1  < z.  <  1.  velocities the  .sum  of  polynomials  representation  Observations  corresponding to the i n f i n i t e  path  space a r e f o r t u n a t e l y  that  occurs  methods There  of t h e C a r t e s i a n i n v e r s e  when  will  not  and v a r i e d  allow accurate interpolation overcome t h e l e a r n i n g  bounds  necessary;  interpolate  must be s u f f i c i e n t  Secondly,  number o f p o i n t s  over  between  observations,  interference  points.  however,  that occurs during  be  imposed  analytical  C a r t e s i a n i n v e r s e dynamics.  A  simple  motion  to  observed  the path  the  as  training.  yet  the  manipulator  h a s e x c e e d e d t h e bounds o f t h i s requirements.  c e n t e r o f p a t h s p a c e i s d e f i n e d by z . e q u a l f  Cartesian  of  < 1 whenever i t i s  The m e t h o d m e e t s t h e a b o v e m e n t i o n e d  This corresponds  Such  unlearned  use  for restoring  s p a c e -1 < z.  t h a t manipulator motion  to  t o be w i t h i n t h e  a n d must n o t r e q u i r e  m e t h o d h a s been f o u n d  be w i t h i n  path space. The  dynamics  using  inverse  t o zero v e l o c i t y  to  zero.  and z e r o a c c e l e r a t i o n w h i l e a t  position,  <x ,,x > = <0,-1> m c 1 c2  (3.161)  0  By  related  t h e whole o f p a t h space and t o  Cartesian  corresponds < a  or  -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 . cannot  in  generalization  training  one must c o n s t r a i n m a n i p u l a t o r m o t i o n  constraints  This  the  using Interference Minimization  effectively  dynamics.  c1' c2 a  >  =  simple s t a t i c  to joint <2  position,  »r/3 ,-TT/3>  (3.162)  a n a l y s i s of t h e m a n i p u l a t o r ,  the corresponding  torques a r e , T , = -8.50 N-m cl  (3.163)  T  (3.164)  ~ — 8.50 N-m c2  121  One  can  apply  manipulator  at  manipulator such  T  2  Since  = T  1  =  r  joint  c  1  joint  position  2  servoed  the C a r t e s i a n path  1  "  j 2 ( a  2  _ a  c2  a c  y  a  2  o  >  c l  space,  restore  r  it.  (3.165) (3.166)  such  simple within  joint  to the center  the  bounds  -1 < z. < 1.  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 these measurements  by  differentiation.  s p a c e o r more c o r r e c t l y  returning the  p a t h s p a c e when i t i s o b s e r v e d  only  necessary  move  so  found  of  s e r v o c o n t r o l c a n be  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  the  the  An e x a m p l e o f  )  t h e a n g u l a r v e l o c i t y c a n be o b t a i n e d f r o m  path  the  i s the following,  j o i n t p o s i t i o n corresponds  angular p o s i t i o n  simple  C  maintain  )  to return the manipulator  The  <  i f d i s t u r b e d from  - j2(a ~a  1  j 1 d  servo c o n t r o l t o  servo control  - j 1 a.  c2 "  the  used  joint  to this position  uncoupled T  uncoupled  within the  manipulator  within  t o be o u t o f b o u n d s ,  i t is  t o f i n d a mechanism t o cause t h e m a n i p u l a t o r  as t o e x p l o r e a l l the r e g i o n s of path space.  t h a t t h i s c a n be done by l e a r n i n g t h e sum o f  We  to have  polynomials  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  partially  manipulator encompass  over the  self-learned the path is  however,  a sequence of randomly generated  regions  of  path  space.  Initially,  paths  the that  when  C a r t e s i a n i n v e r s e dynamics a r e very p o o r l y  the  known,  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  frequently  bounds  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  required to restore the manipulator  -1 < Zj < 1. learning  During  such p o o r l y c o n t r o l l e d  c a n t a k e p l a c e whenever t h e  122  within  the  thrashing,  manipulator  is  within  bounds.  immediately unlearned only  In  become  just  path  learned the  Cartesian  when  dynamics;  instances  inverse  Cartesian of path  inverse  driven  by  learning  when  the may  dynamics  may  as  yet  initially  the constraining  within  dynamics  acceleration  bounds.  Later,  become  more  action  when t h e accurate,  more c l o s e l y .  The  self-  are eventually, learned  space  i f the training  were  performed  paths  cover  over  t h e whole o f  space.  which  self-learning  carried  out.  DELAY=.TRUE., ^ =0.0l  those  used  in  control  higher  feedback  more  The  inverse  instability  Cartesian  Learning  parameters  sec and Af  chapter  Method  algorithm system  used.  order  was  Learning  T h e number s = 4.  The  k =4,  k =4,  1  as  somewhat this  stages  learned  set  as  N«m.  the  Identification,  of input input  123  variables  variables  when  the  create  when  the  learned.  follows:  Using  process;  but  of t r a i n i n g  partially  from  permitted  tracking,  well  and  2  path  a r e only were  follows:  better been  was  VISION=.TRUE.,  reduced  control  as  i n the training  have  . =0.01 min  2,  set  earlier  earlier  dynamics  t, =0.01 lrn1 2,  cause  during  inverse  were  analytical  dynamics  were  during  dynamics  EXACT=.FALSE.,  gains  paths  inverse  USEOBS=.FALSE.,  t o be a c h i e v e d gains  1000 t r a i n i n g  parameters  feedback  standard  of  Cartesian  PRDICT=.TRUE.,  stable  Cartesian  the  INVARM=.FALSE.,  sec.  c  of  Simulation  CLOSED=.TRUE.,  c a  inverse  the manipulator  Simulations  t  manipulator  s p e c i f i c a t i o n s are followed  whole  path  the  of bounds  at those  returned  self-learned the  out  Cartesian  be p o s s i b l e  has  fact  LRNINV=.TRUE., nomenclature was  was  were  of  the learning v = 6 and the  those  given  in  equations  was  (3.138)  through  Constraining  action  was  observed  be  of  uncoupled  to  joint  (3.143).  out  servo  invoked  whenever  bounds.  control  The  defined  the  gains  in  manipulator used  (3.165)  in  and  the  (3.166)  were,  In  j  1  =  4  (3.167)  j  2  =  4  (3.168)  addition  restricted models in  the  the  at  hydraulics, At  a l l times  finite  practice  drive  applied  or  each  within  the  bounds  place.  Then  the  used  observed  <  off-line  to  manipulator By  off-line  not  applied  to  the  of the  the  difference  corresponding  was  not  Cartesian  instant  that  greater  of  t ^  z.  than  <  m  limited  n  ,  1  by  1  learning  torques  were  24  This  N»m.  manipulators; the  was  manipulator  in order of  the  that  motors,  estimate  the  torques  we  mean  that  be  was  the was  the  observed.  for either  124  of  could  inverse  means  time.  In  sum  the  If  this  the  of  dynamics to  were  that  are  self-  self-learning  torque  estimation then  Af,  and  manipulator  torques  the  torques  estimation error,  to  be take  coefficients  this  estimated  applied  to  learned  estimated  that  in real  had  corresponding  these  these  Note  done  motion  partially  Initially  conditionally  learning  Cartesian  that  n  =  in real  the  between  Af ^  TRQLIM  of  to  motion  than  present  i n v e r s e dynamics,  torque the  r  manipulator. have  applied  are  observed  motion.  zero.  the  used.  representation  does  less  coefficients  all  learning  be  torques  the -1  of  capability  interval  First,  were  to  pneumatics  performed.  polynomials  magnitudes  the  is the  at  the  error sum  of  polynomials function  for  Learning  that  done  more and  purposes  assumption  off-line  computer  time  self-learning inverse  is  put  that  time  in  be  closed  one  of  loop  dynamics  adjusted  using  <  one  of  no  knowledge event,  can  i s not  one  of  can  well  as  i t i s reasonable motion  accurately  the  to  temporarily  as  for than  such  devote velocity Since  Cartesian before  assume be  velocity  that  estimates point  assume  self-learning  a  utilized.  b e f o r e , the to  an  self-learning  occurring  and  measured  position.  i t i s reasonable  after,  manipulator  be  Because  in observed  acceleration  Thus  This  contraints,  time  base  to  a c c u r a t e e s t i m a t e s of  changes  position,  paths  were  specification  the  0.1  assumed  seem.  time  capability  can  was  that  purposes  o b s e r v a t i o n s f o r use  in  control.  acceleration between  real  t o work,  made m u c h m o r e  Linear  The  a  question.  observations can  inverse  was  self-learning.  point  is  o b s e r v a t i o n s of  in  torque  i t might  on  a  computing  also  as  based  dynamics  additional  on  Cartesian  motion  to obtaining  from  manipulator  of  without  acceleration  Note  the  corresponding  manipulator  f o r the  unreasonable is  the  of  Identification.  Note exactly  representation  used  for  f o r each  training path  was  purposes. chosen  The  uniformly  limits, a  <  max  velocity  0.9  (3.169)  m/sec  specifications  were  chosen  u n i f o r m l y between  the  limits, 0.05 a  and max  < v, max v  max  <  could  0.95 not  (3.170)  m/sec be  chosen  125  too  close  t o z e r o as  the  path  specification  would  carried  The  out.  uniformly  take final  an  i n o r d i n a t e amount o f t i m e  position  from w i t h i n the  region  of  defined  0.375 < a b s ( x - . , - x  ,) < 0.45 cl 0.375 < a b s ( x . . ~ - x -) < 0.45  each  path  to  was  be  chosen  by,  m  (3.171)  m  (3.172)  111  112  Note  that  position  the of  CZ  final each  p o s i t i o n of each  subsequent  acceleration,  velocity,  specifications  that are  likely  to  path.  and  position  near the  path The  is  the  upper  serve  initial  limits  to  on  avoid  path  b o u n d s of p a t h s p a c e and  thus  i n v o k e c o n s t r a i n i n g a c t i o n e v e n when t r a c k i n g  errors  a r e q u i t e s m a l l . F i g u r e 3.32 shows t h e r a n d o m n e s s o f t h e c h o i c e s of a and v for the first 250 t r a i n i n g paths. The max max restriction the  end  workspace r e s u l t s i n a f a i r l y  within the  t h a t p a t h s b e g i n and  the  w o r k s p a c e by  p o s i t i o n s of  the  the  first  i n a band near the  u n i f o r m c o v e r a g e of  t r a i n i n g paths.  250  Figure  edge  of  positions  3.33  shows  t r a i n i n g paths.  RESULTS  During the is  frequently  at  every  the  takes place  N«m.  of  Later,  quite  i s frequently proportion  which l e a r n i n g takes place  better  of o p p o r t u n i t i e s a t  Finally,  become  as  as  the  manipulator  accurate,  which  l e s s than the  the  126  is  learning Cartesian  proportion  of  decreases since  the  threshold,  of o p p o r t u n i t i e s d u r i n g i s shown i n f i g u r e  place  control  self-learned  at which l e a r n i n g takes place  error The  s e l f - l e a r n i n g , the  thus l e a r n i n g cannot take  ^mi*  proportion  dynamics  estimation  s t a g e s of  o f b o u n d s and  increases.  opportunities  0.01  out  interval  achieved,  inverse  initial  3.34.  A f m  i  each path  = n  at  Figure  3.32  Choices  of  a  paths used i n s e l f - l e a r n i n g  m  a  x  and v  m  a  x  for first  of the C a r t e s i a n  127  250  inverse  training  dynamics  Figure  3.33  View  of  first  l e a r n i n g of the C a r t e s i a n  250 t r a i n i n g p a t h s u s e d  inverse  128  dynamics  in  self-  F i g u r e 3.34 learning  Proportion took  place  of during  learning  opportunities  self-learning  of  at  the  which  Cartesian  i n v e r s e dynamics  As  self-learning  followed errors  more  proceeds,  and more c l o s e l y .  the p a t h  specifications  F i g u r e 3.35  t h a t o c c u r r e d d u r i n g each of the  shows  training  are  the  path  paths.  Path  e r r o r s a r e c l e a r l y b e i n g reduced as t r a i n i n g t a k e s p l a c e . B e t t e r control often.  a l s o means t h a t the m a n i p u l a t o r Figure  3.36  goes out of bounds l e s s  shows the maximum out of bounds  t h a t o c c u r r e d d u r i n g each of the t r a i n i n g p a t h s . and  extent  of  out  of  bounds  training.  129  excursions  are  The  excursions frequency  reduced  with  0.20  0.00  Figure inverse  3.35  Path  0.40  0.60  -io3 TRAINING PATHS  errors during  0.80  self-learning  dynamics  130  of the  Cartesian  o  CM '"" -  CJ UJ CO  •-V.O  OO X o  <LkiL.k 0.00  J<| . . . . f l .  0.20  0.20  .1.  I i  A  •IO*  j  ill  t |  I  ,  I  i  ,1  1  Li  0.40  0.60  0.80  1.00  0.40  0.60  0.80  1.00  TRAINING PATHS  •103 T R A I N I N G P A T H S  o  in  CD O •JULJ  0.00  F i g u r e 3.36  .  —  0.20  Maximum  1  I  •io3  0.40  1—•  0.60  TRAINING PATHS  out  of  bounds  —  i  0.80  excursions  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  131  1  1.00  during  self-  It  i s acceleration that  most f r e q u e n t l y  out of bounds.  i s most p o o r l y In fact,  tracked  slight  and  hence  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 c a n e x c e e d t h e b o u n d s on a c c e l e r a t i o n cause in  constraining  velocity  because  action  t o be i n v o k e d w h i c h , may t h e n  and p o s i t i o n e x c e e d i n g t h e i r  constraining  action  simply  within  bounds w i t h o u t c o n c e r n a s t o  action  i s ceased,  to  the large  returns where.  result  This  the When  path tracking error  occurs  manipulator constraining  that  i s likely  e x i s t may c a u s e 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 w h i c h  t a k e s t h e m a n i p u l a t o r o u t o f bounds. of  bounds.  and  Thus a s l i g h t  again  overshooting  s t e p s i n a c c e l e r a t i o n c a n l e a d t o an e p i s o d e o f t h r a s h i n g  which the  velocity out  and p o s i t i o n bounds a r e a l s o e x c e e d e d .  o f bounds e x c u r s i o n s  that  occurred during  in  Many  the  latter  s t a g e s o f t r a i n i n g a p p e a r e d t o be o f t h i s n a t u r e . C o n s t r a i n t s acceleration learning. however,  are  necessary during  Once t h e C a r t e s i a n  the i n i t i a l  inverse  stages of  dynamics a r e w e l l  i t a p p e a r s t h a t more r e l i a b l e c o n t r o l  is  the l i m i t a t i o n A  better  constraining motion. then of  After  of a p p l i e d constraining  action  that  observed  to  learned, achievable  training  path  could  manipulator position.  acceleration.  might  be  to  damp o u t  i s stopped,  relies  joint  invoke  manipulator  servoing  could  move t h e m a n i p u l a t o r t o w a r d s t h e  path space, again  technique  would i n i t i a l l y  the manipulator  be u s e d t o s l o w l y Cartesian  torques t o l i m i t  as before.  Once t h e  be w i t h i n C a r t e s i a n be This  begun,  starting  should avoid  132  path  on  self-  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 i m p o s e d ; one t h e n on  of  center  manipulator  space, from  the  the  the protracted  is next  current episodes  of  thrashing  technique As  described  dynamics  2948  then  INVERSE  the  simplistic  required  s e l f - l e a r n i n g of  1000  training  (49 m i n u t e s )  LOOP  paths.  paths.  were  CONTROL U S I N G  saved  paths  self-learned Simulation  of the  after  Simulations  standard  of  were  Cartesian  THE  would  PRDICT=.TRUE.,  corresponded  If t ^ also  r  n  were  1  increase.  SELF-LEARNED  self-learned  200,  400,  closed  loop  CARTESIAN  are  shown  control  were  dynamics set  3.37  as  follows:  ^=16,  k  over  function  =  6  4  a  Figure  3.38  shows  simulations of  of l i n e  the simulation  Cartesian  inverse  i n appendix  of c i r c l e  dynamics.  133  d  t  the  calc °*° =  Plots C.  1 using  of  inverse  1 using  dynamics.  1 a r e shown  of  1  the s i x standard  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  inverse  six  DELAY=.TRUE.,  o f t h e number  of l i n e  the  CLOSED=.TRUE.,  2  used  the simulation  n  training  progressed.  VISION=.TRUE.,  averaged  as a  as  inverse  1000  of a l l of  paths  Cartesian  and  learning  errors,  in figure  6 0 0 , 800  inverse  USEOBS=.TRUE.,  r e s u l t i n g path  Cartesian  the adequacy  EXACT=.FALSE.,  The  learned  Cartesian  c a r r i e d out t o t e s t  parameters  INVARM=.FALSE.,  plot  This  of s e l f - l e a r n i n g .  s e l f - l e a r n i n g time  coefficients  dynamics  a  the  DYNAMICS  The  learned  constraining  previously.  required  CLOSED  using  previously,  seconds  increased 3.8.2  occurred  mentioned  inverse to  that  Figure the  S  C  *  paths, training  dynamics.  the f i n a l of  S  self-  the  other  3.39  shows  final  self-  F i g u r e 3.37 the  Average  s i x standard  path e r r o r s during closed loop c o n t r o l  p a t h s 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 as a f u n c t i o n of the number of t r a i n i n g p a t h s used.  134  of  4.00  4.00  T SEC  F i g u r e 3.38  C l o s e d l o o p c o n t r o l of l i n 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  135  4.00  1 u s i n g the f i n a l  self-  Figure  3.39  self-learned  Closed  loop  Cartesian  control  inverse  of c i r c l e  dynamics  136  1 using  the  final  Torque dynamics  estimates  were  compared  using  to torque  Cartesian  i n v e r s e dynamics.  standard  paths  number  Path until  errors  minimum  paths  appears  that  control  performance  the  limit  sum  of  o f what  error  errors after  with  the  averaged as a  decrease After  cannot 1000  inverse  analytical over  function  be  with  a  r e p r e s e n t a t i o n of  during  1000  the s i x of  the  paths 4 ^ fc  self-learning  training  reduced  training  c a n be a c h i e v e d  polynomials  3.40  Cartesian  used.  are reached.  these  error,  in figure  and torque  values  calculated  Torque  a r e shown  of t r a i n i n g  the s e l f - l e a r n e d  further.  thus  order,  the  paths  i t The  represents self-learned  Cartesian  inverse  dynamics. The  resulting  standard over in  paths  1000 table  applied poorer errors  using  training 3.15 during  than  path  each  with  the torque standard  less  i n v e r s e dynamics  are listed  standard  are typically  f o r the s i m u l a t i o n s of  the Cartesian  paths  are  errors  in table  path.  analytical than  3  mm.  137  six  self-learned Also  listed  the  maximum  torque  Performance  i s only  slightly  Position  tracking  error  and  3.15.  the  control.  -  LU  CM  ~*0.00  0.20  0.40  0.60  0.80  1.00  -io3 TRAINING PATHS  F i g u r e 3.40  Average  control  the  Cartesian  of  torque estimation e r r o r during closed  six  inverse  standard  dynamics  paths  using  as a f u n c t i o n  t r a i n i n g p a t h s used.  138  the of  the  loop  self-learned number  of  PATH  AXERMS  TERMS  TRQMAX  1  1 .60E-3  4.02E-3  1.01E-l  1.96E-1  8.27  line  2  2 .15E-3  4.75E-3  1.03E-1  2.47E-1  9.41  line  3  1 .47E-3  3.90E-3  1.08E-1  1.64E-1  14.0  line  4  1 .74E-2  4.09E-2  2.92E-1  3.00E-1  6.64  circle  1  1 .16E-3  3.85E-3  5.29E-2  1.52E-1  9.59  circle  2  1 .62E-3  3.45E-3  5.26E-2  2.00E-1  13.5  3.15  learned  Path  error  and torque  error  C a r t e s i a n i n v e r s e dynamics  S E L F - L E A R N I N G OF T H E D I R E C T  A  sum o f p o l y n o m i a l s  kinematics control.  has  observation  We  only  A  POSITION  to find  without  have  end p o i n t  the vision  i n closed  the  loop  final  found  KINEMATICS  recourse  joint  position  adequate  this  positions  to  position  for  a method of l e a r n i n g  that  self-  control  representation of the d i r e c t  of manipulator  manipulator  using  p r e - l e a r n e d and found  kinematics  manipulator.  3.9.1  been  I t remains  position  by  VXERMS  line  Table  3.9  XERMS  the direct  analysis  can  be  coarse  done  of  the  through  and the corresponding  i n Cartesian coordinates as  seen  system.  METHOD  FOR  SELF-LEARNING  OF  THE  DIRECT  POSITION  KINEMATICS THE  METHOD Given  position  sufficient in  joint  and v a r i e d  observations of the manipulator  coordinates  139  and  manipulator  position  in  Cartesian related  methods  To position two  coordinates,  the  space  given  conditions  z.  direct  in  Since over  the  direct whole  the  the  that,  as  of  position  space,  of  the  the  i t  i s only  done  Simulations  one  bounds  -1  <  -1  z.  to  1.  direct  position  by  work  -1  the  z.  <  act  <  as  1.  input the  constrain  the  <  in  order  kinematics  only  z  <  given  by  -1  space  1  <  i  constrain was  position  1.  thus  140  of  4200  since  the  (3.151)  within the  possible  inverse  performed  the  to  within perform  simultaneously dynamics.  were  taking  the  Cartesian  manipulator  self-learned Cartesian  s e l f - l e a r n i n g was  over  Also,  kinematics  kinematics  the  not  motion  the  learn  equations  inverse  position  we  s e l f - l e a r n i n g of  It  the  that  Cartesian  were  of  of  z.  space,  manipulator  during  serves  while  same  observations  that  must  necessary  the  given  joint  <  direct  over  also  direct  purposes  the  the  representation  Secondly,  use  the  the  positions  joint  place.  s e l f - l e a r n i n g of  self-learned  obtain  within  space  <  (3.152),  must  joint  the  self-learning  polynomials  kinematics  was  over  one  the  or  kinematics.  kinematics  during  Minimization  First,  constraining  space  unlike  control  to  is within  self-learning with  be  take  position  dynamics  joint  to  wish  (3.152),  inverse  of  workspace,  joint  workspace  sum  position  (3.151) and  as  kinematics.  can  position  workspace and  we  met  normalized  the  motion  learning  equations be  Interference  position  dynamics:  the  use  direct  a l l regions  position  manipulator that  to are  variables  by  can  the  direct  must  inverse  corresponding the  learn  learn  Cartesian  Here  to  one  Note  dynamics,  the  not  for  used  place.  training  paths  during  which  self-learning  carried the  out.  For  Cartesian  Thus  the  Nothing  Learning t, ,,=0.01 lrn2 Minimization and  the  given  in At  be  take of  was  interval  the  place. polynomials used  the  were  between  the  <  the  error  joint was  of  the  coordinates  r  n  direct  position  of  given  the  the  final  Cartesian  LRNINV=.FALSE. LRNVIS=.TRUE., 3,  Interference  input  variables  was  The  input  variables  were  2 '  z.  learning  was  manipulator <  1  of  in  order  of  the  the  estimate  zero.  v  =  2  those  that  positions learning  partially  positions that  the  were Af  .  sum  kinematics  Cartesian  position  positions.  Af,  p o s i t i o n and by  could  learned  s e l f - l e a r n i n g of error,  had  position  joint  In  conditionally  joint  direct the  observed  greater  then  out.  as  simulation  Method  i s the the  the  direct  difference  corresponding  vision  system  observed. for  These  either  at  If of  the this the  mm  3  Cartesian  same  of  estimation  was  carried  (3.152).  Cartesian  that  was  s e l f - l e a r n i n g of  follows:  m.  observed  the  estimated  position  estimation  to  initially  kinematics,  that  i  as  coefficients  the  position  instant  t  the  set  4.  representation  to  coefficients  Cartesian  -1  off-line  corresponding  =  the  setting  observed  bounds  Then  by  number  s  of  the  were  . =0.001 nun  was  paths  s e l f - l e a r n i n g of  were  The  kinematics  simultaneously  for  that  (3.151) and  First,  within  were  order  equations each  Af  used.  system  performed. to  and  was  changed  disabled  position  training  parameters  parameters  sec  1000  except  was  direct  dynamics  was  paths  dynamics  the  first  inverse  training  inverse  the  simulation  previously. 3200  of  the  sum  kinematics  141  of  polynomials  function  for  the  representation corresponding  Cartesian Minimizat The  coordinate  was  using  adjusted  Interference  ion. .training  described  paths  previously  that  for  were  used  s e l f - l e a r n i n g of  were the  the  same  Cartesian  as  inverse  dynamics. RESULTS  The  proportion  learning  takes  self-learning  of  place  of  the  opportunities is  Cartesian  remains  relatively  position  kinematics.  This  exceeds  the  on  positions  than  i t  not  estimation Af  . mm  =  increased, training kinematics required. 0.01  m  did  The kinematics  the  be  error  to  Training  4 ^  0.01  m,  the  at  where  points speed  saved  of at  Af  proportion the  direct  representing  normalized  on  less  input positions learned the  system  thus  less Af  and  than . mm  of  min  the  position  been  number  direct  between  position  have  learning  was  the  threshold,  could  the  for  workspace  the  increasing  often  variables  throughout  number  during  manipulator  point  that  which  the  self-learn  reducing  to  m  without to  this  that  typical  order  fc  c  coefficients were  0.001  at  Unlike  kinematics,  end  Also,  i s probable  necessary  while  little  a  fact  bounds  infrequently  It  say  paths  extensive  with  was  m.  the  position  than  3.41.  path  s e l f - l e a r n i n g of  Cartesian  less  each  dynamics,  variables  direct  more  achieved  0.001  inverse  dynamics. of  figure  reflects  normalized  error  in  during  input  the  inverse  estimation could  for  exceeds  representing Cartesian  constant  bounds  joint  shown  during  of  position iterations  0.001  m  and  convergence. the various  142  self-learned points  direct  during  the  position learning  process.  Simulations  analytical  of c i r c l e  control  to  specification.  The  used  a view  to obtain  Figure  3.42  estimation  self-learning shows  the  position direct are path  of c i r c l e  position  kinematics  i n appendix  initially  not improving  after  4200  after  training  accuracy  that  sum  polynomials  of  kinematics. the  final  of the view  I t c a n be become 4200  paths  c a n be a c h i e v e d  Table  lists  direct  used  Figure  self-learned the  path in  3.43 direct  self-learned  s i m u l a t i o n s of c i r c l e that  1  accurate  with  training  paths.  The  a  the path  143  of  paths  self-learned, of  the  direct  the 4 ^ fc  best order  position  estimation errors kinematics.  but  performance  r e p r e s e n t a t i v e of  position  were  self-learned  representation  3.16  self-learned  using  path  the  training  i s thus  accuracy  using  seen  more  the  simulations.  kinematics.  the f i n a l  standard  kinematics  of t r a i n i n g  f o r the other  D.  of  of the  in the  position  1 using  Plots  position  1 f o r each  improvement  out using  tracking  direct  f u n c t i o n o f t h e number  kinematic^..  estimates  are  the  carried  accurate  o'f c i r c l e  of the d i r e c t  view  shown  ensure  self-learned  shows  as a  1 were  using  1.00  F i g u r e 3.41 learning  (i\20  Proportion  CL60  (MO  -io3 TRAINING PATHS  of  learning  euro  opportunities  1.00  at  which  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 of t h e d i r e c t p o s i t i o n  kinematics  144  Figure  3.42  Path estimation e r r o r s during  direct  position  kinematics  145  self-learning  of  the  ow o •  o O' i  Figure  3.43  View  direct position  of  circle  1 using  kinematics  146  the  final  self-learned  Table  PATH  LXERMS  LVEPvMS  LAERMS  line 1  1.37E-1  1.66E-1  4/17E-1  line 2  9.96E-2  1.13E-1  2.90E-1  line 3  2.75E-2  4.65E-.2  1.64E-1  1 ine 4  7.56E-2  7.86E-2  2.52E-1  circle 1  4.97E-2  9.61E-2  3.1OE-1  circle 2  5.28E-2  9.01E-2  3.02E-1  3.16  direct  Path  position  Table  estimation error  using the f i n a l  self-learned  kinematics  3.17  lists  the coefficients  for  the f i n a l  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 d i r e c t kinematics  alongside  correspondence The  outside  The  f o r the higher order  terms.  i n t h e h i g h e r order terms i s n o t s u r p r i s i n g  direct  position  t h e b o u n d s on z^. f o r  kinematics;  t h e workspace and thus l i t t l e  these  self-learaing  as the  bounds a r e took  place  them. Self-learning  of  the direct  position  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 24  position  coefficients.  t e r m s a r e most s i g n i f i c a n t n e a r  self-learned  near  derived  i s q u i t e good e x c e p t  innaccuracy  these  the  self-  minutes)  of  self-learning.  147  kinematics  t o 12250 s e c o n d s  required (3 h o u r s  POLYNOMIAL TERM  X(1) COEFFICIENTS DERIVED SELF-LEARNED  1  0.0  z  1  Z  1  1 .57 2  «,  0.0 -0.646  3  -I* Z  2  Z  2 1 Z  2  Z  z  Z  Z  2  3  0.0  0. 155  1 .36  1 .58  1 .23  1 .62  -2.60 -0.276 1 .49  -1 .68  -0.750  -0.507  -1.41  -1 .07  -0.879  0.617  0.277  -0.969  -0.051  -0.839  -0.699  1 .32  -0.492  -0.761  -2.05  -0.323  -0.653  -0.559  -1 .54  0.879  1 .45  -0.507  -0.633  «2*  0.220  0.982  -0.127  1.13  3  sum  -1 .77  z z 2 1 z  Table  2.20  0.787  0.879  z z 2 1 2 2 2 1  1 .23  0.785  3  z  -0.006  -0.040  -2.14  Z  2 "  0.0  -0.731  z z 2 1 Z  1 .67  -1 .04  -0.254  -0.969  Z  -1 .00  0.209  2  Z  -0.009  0.0  z z 2 1 Z  X(2) COEFFICIENTS DERIVED SELF-LEARNED  Z  3.17 of  Coefficients  polynomials  kinematics  functions  f o r the derived  representations for positions  148  x.  of and  and the x«  final  self-learned  direct  position  From * t h e inverse  derivation  dynamics  expect  the  Several  factors  direct  self-learn position to  do  the  and  a  position  likewise  .  correction  inverse  to to  be  sign  known.  required  manipulator.  in  providing  the  direct  inverse  dynamics,  six  input  together  time  direct  by  these  being  This  means  are often  the s i x input  position  When  kinematics  variables  to  only  be  be  known  only  t o be  kinematics  must  self-learned  into  tracking  and,  of system  non-  control  are less  that  i n changes  Furthermore, training i n terms When  are only  quickly.  input  of  Cartesian to a l l of steps  in  close  the  space  self-learning  two  of  effective  points of  be  direct  errors  acceleration  result  variables.  149  error  s e l f - l e a r n i n g the  used.  a r e not changing  very  f o r the s e l f - l e a r n i n g  f a r apart  there  than  feedback  ST^./SCL ,  the t r a i n i n g paths  kinematics.  direct  known  needs  to  First,  the  resolved  learn.  purposes  be  the cancellation  training points  variables  occur.  represented  obviate  the t r a i n i n g paths  acceleration in  in  to  the  i t appears  position  directly  f o r stable  suitable  For  the magnitude  the d i r e c t  would  dynamics. to  differential,  Errors  Finally,  position  do not need  space;  translate  s i g n i f i c a n t enough,  the  and  path  known.  kinematics  linearities  the  of the  Secondly,  accurately  inverse  effective  one  of  for control  reasonably  throughout  if  are adequate  Cartesian  i t more d i f f i c u l t  representation  control.  accurately  position  t o make  good  that  very  t o be e a s i e r  dynamics  achieve  necessary  roughly  kinematics  f o r the the Cartesian  Cartesian  the  kinematics,  however,  that  of  position  of polynomials  kinematics  accurately  direct  combine, sum  and p r e - l e a r n i n g  the  variables  Training  points  close  together  space  in  represented  time by  are  the  Minimization  eliminates  point  and  the  sum  will  be  little  points. occur  position  seems  training  only  kinematics  are  at  the  not  a l l  occur  one  is  place time  terms  of  the  Interference  at  a  training  continuous, subsequent  when  to  there training  estimation  self-learn  of  accuracy  the  errors  the  direct  that  smoothing introducing  a  of  estimation  errors  the  can  the in  are  i s due  to  the  region  of  workspace  points  are  less  achieve final of  stages  less  reduced  150  of 1  final  favor in  a  self-  therefore stages  of  averaging  or  self-learning  by  in  Minimization at  not  and  in  during  modify  similar  than  Interference only  a  to  the  training  results  frequent  during  which  points  thus  final  points  does  is possible  workspace  the  and  last  other  training  in  self-learned  the  at  the  path  the  of  inaccuracy  It  difference  training Thus  direct  pre-learned  self-learning,  space. the  the  position kinematics  factor for  as  final  choosing  also  during  gain  formula  the  is better.  One  adjustment  In  of  direct  over  effect  chosen.  random  training  distributed  this  on  self-learned  accurate  increased  region  that  self-learning.  of  as  the  Again,  accurate  The  particular  so  region  quite  workspace.  learning more  are  are  expense  pre-learning  mean  exactly  nearby  s u r p r i s i n g that  points  points  represents  one  more  position kinematics.  way  at  takes  i t requires  kinematics  training  the  only  error  estimate  error  in  v a r i a b l e s . Since  estimation  estimation  hence  also  position  path  input  together  kinematics.  It  the  two  close  polynomials  Self-learning and  direct  of  thus  training  the  weight  such points,  that not  eliminated. final  These  stages  convergence To  learning  and  1001  changed  the  to  adjustment  t  i  n  1  formula  Minimization  by  1  would  for  averaging  were  simulated  paths  s  position  except e  was  •  c  I  of  a  from gain  the  the  addition  n  the  the  rate  of  smoothing,  an  self-learning.  were  that  modified  inclusion  of  during  reduce  or  during  kinematics  parameters  4200, 2 ^ '  applied  methods  direct  =  r  they  be  stages  learning  to  as  only  early  training  of.  Simulation  the  these  400  should  self-learning  during  test  additional  paths  of  methods  was  same  that  as  for  out.  training  interval  was  learning  algorithm  normal  Interference  of  factor  self-  carried  learning the  which  of  1/10  as  follows,  AfP~ p -= =1  Aw  =  1/10  P P After were  completion  carried  Standard  out  paths  cations.  The  and  used  Table  lists  the  Figure  3.44  Table learned As  lists  order  Clearly,  the  to  a  view  of  of  errors  view  coefficients  kinematics  after  correspondence  improvement.  simulate the  for each  circle  for  position  path  path. using  kinematics. 1.  the  final  additional  i s quite  six  specifi-  standard  position of  the  path  direct  each  direct  simulations  of  self-learned  corresponding  position  extent  tracking  estimation  the  smoothing,  used  self-learned  3.19  3.17,  obtain  path  the  direct  was  final  to  shows  in table  higher  final  the  accurate  smoothed,  smoothed,  additional  determine  ensure  were  the  this  control  kinematics 3.18  of  to  analytical  standard  (3.173)  P  good  self-  smoothing. except  for  smoothing  has  terms. the  additional  self-learning  151  with  improved of  the accuracy  the d i r e c t  additional further  such  position  modifications  improvements.  order,  of polynomials  position kinematics.  improvements 6  o f t h e sum  i s t o use a higher that  can  kinematics,  more without  It i s doubtful,  of  Probably  representation  self-learning a  order  though, could  yield  better  way  sum  polynomials,  accurately requiring  of  to  represent special  that  achieve  the  say direct  variations  of  self-learning.  PATH  Table  LXERMS  LVERMS  LAERMS  line  1  5.32E-2  7.43E-2  2.73E-1  line  2  3.08E-2  4.42E-2  1.77E-1  line  3  1.16E-2  2.37E-2  9.48E-2  line  4  2.33E-2  3.37E-2  1.41E-1  circle  1  1.04E-2  3.48E-2  1.47E-1  circle  2  1.17E-2  3.42E-2  1.45E-1  3.18  self-learned  Path  estimation  direct  error  using  position,kinematics  152  the  smoothed,  final  POLYNOMIAL TERM  X( 1) C O E F F I C I E N T S DERIVED SELF-LEARNED & SMOOTHED  1 z  1  Z  1  z  1  Z  1  Z  2  Z  2 1  Z  2 1  2  Z  2 Z  z z 2 1 2 2  3  Z  Z  z z 2 1 2 2 2 1 2  z  Z  z  Z  Z  2  3  z z 2 1 4 2 3  Z  Z  z  Table  3.19  self-learned position  0.009  -1 .00  -0.993  1 .57  1 .64  0.0  -0.015  -0.646  4  Z  0.0  0.0  3  -0.186  1 .23  1-11  -1.61  0.0  0.234  0.0  0.345  -0.254  -0.437  0.785  0.809  1 .36  1 .39  1 .23  1 .28  -2.14  -2.34  -0.969  -0.453  0.879  1 .36  -1 .07  -1 .26  -0.969  -1 .68  -0.799  -0.507  -1.13  0.617  0.407  -0.076  -0.839  -0.794  1 .32  -0.312  -0.761  -1 .84  -0.323  -0.568  -0.559  -1.21  0.879  1 .21  -0.507  -0.383  0.220  1 .38  -0.127  1 .00  Coefficients  for  the derived  sum o f p o l y n o m i a l s  kinematics  X(2) COEFFICIENTS DERIVED SELF-LEARNED & SMOOTHED  functions  and  smoothed,  representations  f o rpositions  153  x  1  of  a n d x~  the  final direct  Figure  3.44  View  learned direct  of c i r c l e  position  1 u s i n g t h e smoothed,  kinematics  154  final  self-  3.9.2  CLOSED  INVERSE  LOOP  DYNAMICS  AND  Simulations using  the  learned  CONTROL U S I N G THE  SELF-LEARNED  verified  self-learned  direct  Simulation  position  that  Cartesian  were  set  PRDICT=.TRUE.,  We  standard  high  as  obtained  they by  gains,  resulted  k =8,  A l l s i x standard  errors  a r e shown  Tracking direct  self-learned  paths  in table  i s worse  kinematics,  pre-learned  the  same  order  shown path,  in as  too  results  to the The  position  kinematics  position  kinematics.  Figure  final  path  3.45  shows  1.  system  i s unable  t o make  i s somewhat  expected.  155  errors  well. worse  the  Control  than are of  the  the  as  resulting position is clearly perceived  The a c t u a l than  final  errors,  direct  of c i r c l e  and  as the  estimation  control  that  pre-learned  accurate  Path  self-learned  specification  3.46  given  resulting  the  are less  as p o s i t i o n  3.16.  the  blind  were  values  expected  as the c o n t r o l  figure  were deemed  best  simulated.  control.  sec.  i s t o be  using  follow the path  gains  self-  , =0.01 C3 XC  this  as  path  The  t  however,  path  marginal  and  2  and  3.20.  of magnitude  during  0  possible  DELAY=.TRUE.,  with  direct  is  CLOSED=.TRUE.,  achieved  previously in table  kinematics  k =16 Z  and k =64,  1  were  KINEMATICS  blind  that  shown  seen  this  than  direct  the  k =16  the feedback  above.  call  in instability.  reducing  control  VI SION=.FALSE.,  USEOBS=.FALSE.,  feedback  POSITION  follows:  1  The  loop  CARTESIAN  inverse dynamics  as  EXACT=.FALSE.,  SELF-LEARNED  DIRECT  closed  kinematics.  parameters  INVARM=.FALSE.,  THE  path  i s  perceived  I  o rs. o o O) O' I  CNJ—'  © in ^0.50  -0.30  F i g u r e 3.45  View  -0.10  LEX(1)  of  circle  M  0.10  1 using  0.30  the  final  0.50  self-learned  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 1  using  circle  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 dynamics  final self-learned direct position  156  kinematics  and  F i g u r e 3.46 circle  View  of  circle  1 during closed  loop  control  of  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 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  157  kinematics  PATH  Table  AXERMS  1  1.37E-1  1.66E-1  4.17E-1  line  2  9.96E-2  1.13E-1  2.90E-1  line  3  2.75E-2  4.65E-2  1.64E-1  line  4  7.56E-2  7.86E-2  2.52E-1  circle  1  4.97E-2  9.61E-2  3.10E-1  circle  2  5.28E-2  9.01E-2  3.02E-1  3.20  Path  standard listed  direct  paths  were  in table  3.21.  Tracking learned  i s  direct  additional  position  magnitude  as p o s i t i o n  smoothed,  during  blind  Figure  The  final control  out using  position  not  achieved  path  with  This  has rendered  t h e smoothed,  errors  estimation shows  of c i r c l e  i s t o be  are of the  the resulting  1.  158  errors  are  the  pre-  expected final  same  as given path  loop  A l l six  as the pre-learned  errors,  self-learned  smoothed,  altered.  resulting  as that  the  in closed  kinematics.  Path  3.47  Cartesian  direct  kinematics  were  t o be a s a c c u r a t e  kinematics.  3.18.  positions  as good  averaging  position  table  were c a r r i e d  simulated.  position  kinematics  self-learned  parameters  now  self-learned  control  simulations  Simulation  the f i n a l  final  loop  self-learned  control.  using  and  in closed  Additional final  error  dynamics  kinematics  the  VXERMS  line  inverse  the  XERMS  direct direct  order  previously as seen  direct  position  Control  i s much  as  of in  using  kinematics better  now  than  when  kinematics to  make  actual the  using without  path  final  smoothing,  the perceived i s shown  perceived  path,  PATH  Table  the  path  self-learned since  follow  in figure  as  direct  the control  the path  3.48  position  system  i s  specification.  a n d i s somewhat  worse  XERMS  VXERMS  AXERMS  1  3.83E-2  9.27E-2  4.04E-1  line  2  8.41E-2  1.18E-2  2.34E+0  line  3  1.15E-2  2.71E-2  1.41E-1  line  4  1.85E-2  3.53E-2  1.65E-1  circle  1  1.11E-2  3.86E-2  1.71E-1  circle  2  1.40E-2  4.13E-2  1.79E-1  inverse position  Path  error  dynamics kinematics  and  The than  expected.  line  3.21  able  using  the f i n a l  smoothed,  in closed  loop  159  self-learned  final control  self-learned  Cartesian direct  o in  o  o CO  (\J— .  *  x7 LU  o CO  o in 0.50  Figure learned circle and  -0.30  3.47  View  -0.10  LEX(i) M  of c i r c l e  direct position 1 using  smoothed,  the f i n a l final  0.10  1 using  t h e smoothed,  kinematics during closed self-learned  self-learned  Cartesian  direct position  160  0.50  0.30  final  self-  loop control inverse  of  dynamics  kinematics  o  in &  on  o fx  O ' I  Figure  3.48  circle  1 using  and  smoothed,  View  of  circle  the f i n a l final  1 during  self-learned  self-learned  closed  loop  Cartesian  direct position  161  control  inverse  of  dynamics  kinematics  Figure of  line  to  1.  the  The  from  smoothed,  when  glitch  iteration  smoothed, 3.50  shows  where  this  before  final  the  technique  that  to the  position  direct  path  during  enabled  by  162  blind  setting  a r e due  when  less  direct by  As  an  Interference  system  kinematics. control  was  position  utilizing  or  one  accurate  kinematics.  the vision position  control  occur  results  pre-learned  from  blind  initially  Identification  switching  was  during  be e l i m i n a t e d  of Learning  self-learned  system  direct  can  the resulting  error  vision  using  path  that  position  self-learned  this  Minimization  accelerations  the accurate  before  initial  the r e s u l t i n g  perceived  final  kinematics,  shows  large  sudden  switches  done  3.49  of  ADJVIS=.TRUE.  to  the  Figure line  1  F i g u r e 3.49 self-learned  Closed  loop  Cartesian  c o n t r o l of l i n e 1 i n v e r s e dynamics  self-learned direct position  kinematics  163  and  using  the  final  smoothed,  final  4.00  '0.00  0.80  1.60  2.40  3.20  4.00  T SEC in  in  4.00  T SEC  F i g u r e 3.50 self-learned  Closed  loop  Cartesian  c o n t r o l of l i n e 1 i n v e r s e dynamics  using  and  the  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 w i t h ADJVIS=.TRUE.  164  final final  These using  simulations  the  learned  show  that coarse  control  self-learned Cartesian inverse  direct  position  kinematics.  dynamics  position  techniques  kinematics.  during  the  possible  to  position  kinematics  better  is  possible  direct  position  higher  order  more  over  to  using the  kinematics.  workspace.  Since  sum o f p o l y n o m i a l s  computational  costs  or  i t is direct  equal  to  kinematics. final  or  Good  self-learned  improvements a r e l i k e l y i f  representations are the d i r e c t  position  used  that  kinematics  t h e r e a r e o n l y two i n p u t  variables  k i n e m a t i c s , one c a n u s e h i g h  representations without  of having  to  smoothing  self-learned  position  smoothed,  when e s t i m a t i n g t h e d i r e c t p o s i t i o n order,  the  Further  sum o f p o l y n o m i a l s  self-  self-learning,  the p o i n t that they a r e  accurately represent  the  of  of  than the p r e - l e a r n e d d i r e c t  control  can  stages  improve t h e accuracy  and  of t h e s e l f - l e a r n e d  By u s i n g a v e r a g i n g  final  possible  F i n e r c o n t r o l appears  r e q u i r e o n l y an i m p r o v e m e n t i n t h e a c c u r a c y direct  is  i n c u r r i n g the  a l a r g e number o f t e r m s  i n the  sums.  3.10 SUMMARY The its  c o n c e p t o f s e l f - l e a r n e d c o n t r o l h a s been i n t r o d u c e d a n d  performance  demonstrated  manipulator.  This  performance  achieved  a  simulated  two  link  p e r f o r m a n c e h a s been c o m p a r e d t o t h e c o n t r o l using  two l i n k  the  kinematics  of  analytical  dynamics and k i n e m a t i c s  to  that standard  ensure  the  using  analytical  manipulator.  Control  and  using  h a s been e x t e n s i v e l y  a n a l y t i c a l c o n t r o l forms  165  dynamics  a  the  tested  realistic  benchmark a g a i n s t w h i c h t o compare Self-learning represent  the n o n - l i n e a r ,  relationships  algorithms, the  of  relationships  recourse  learn  as  manipulator's  to learn  the weighting thus  a  as  functional  thought  of as merely series  coefficients  the  The sum o f p o l y n o m i a l s  i t  is  without  the  can  be  whole  replacing  derivation  the v a r i o u s sine  of  representation  by j u s t  representation  t h u s y i e l d s an e f f i c i e n t  a  such a t e c h n i q u e .  of the f u n c t i o n a l these  in  a  representation  e q u i v a l e n t s and c o m b i n i n g  of  functional  relationships  i n v e r s e dynamics appear  by  learning  learning  well  of  the  T h i s would seem t o be t r u e i n g e n e r a l f o r  demonstrated  evaluation  of  of the m a n i p u l a t o r .  their  [28,46].  to  sums  Furthermore,  sum o f p o l y n o m i a l s o v e r  format  the  learn  to achieve c o n t r o l .  these  reach.  manipulators  their  as weighted  i n v e r s e ^ d y n a m i c s o f t h e two m a n i p u l a t o r  represented  by  i t i s possible  dynamics and k i n e m a t i c s  manipualator  p o l y n o m i a l s and  to analysis  The  control.  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  necessary  to  that  multi-variate  i t i s possible  sums  possible  on t h e f a c t  o f t h e two l i n k  polynomials. Using  of  i s based  self-learned  similar can  and c o s i n e  alike  terms.  sum  of  be  terms  T h i s was  polynomials  The sum o f p o l y n o m i a l s computational  relationships  as w e l l  relationships  with  format f o r  as p e r m i t t i n g Interference  Minimization. The be  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  represented  manipulator's manipulators  by sums o f p o l y n o m i a l s o v e r reach  at  a  time.  The  i n general are d i f f i c u l t  166  manipulator a  inverse  portion  can only of  kinematics  the of  t o o b t a i n a n a l y t i c a l l y and  are  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  by  sums o f p o l y n o m i a l s o v e r  Nevertheless link  we  found  manipulator  over  a  represented  t h e whole of a m a n i p u l a t o r ' s  that the inverse kinematics of  c a n be w e l l  workspace  be  reach.  the  two  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  consisting  of  a  sizable  portion  of  the  manipulator's  r e a c h . Hence 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 c a n be  self-learned  over  specially  chosen;  could  have  whole  manipulator  of  this  workspace.  other  been u s e d .  The  p o r t i o n s of the  workspace  I t w o u l d be p o s s i b l e t o  solution learned  One  sum  splitting 5.  Another  t h a t may be p o s s i b l e i s t o u s e a more g e n e r a l t y p e  consisting  of  that  p e r h a p s b a s e d on  a r a t i o o f two sums  of  s e l f - l e a r n i n g c a n be a c h i e v e d  whole of t h e i r  reach  i fa splitting  control  method  within  inverse  linearities  of  system  i s approximately  which  the  dynamics manipulator  simple error c o r r e c t i n g Self-learned coordinates  a r e used  these  to  in over  o r o t h e r more used. self-learned  relationships are applied i s resolved acceleration self-learned  would  r e a c h and  technique  which  It  f o r manipulators  g e n e r a l t y p e o f 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 c a n be The  of  representation  polynomials.  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 the  the  unique  i s discussed i n chapter  functional estimation,  appear  reach  subdivide  space i n t o p o r t i o n s and s e l f - l e a r n  t h a t may be v i a b l e  not  manipulator's  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 .  technique  was  control.  cancel  dynamics y i e l d i n g  the a  The non-  combined  l i n e a r a n d c a n be c o n t r o l l e d  using  feedback.  control  using path  w o u l d a p p e a r t o be e a s y  167  specification  t o apply  in  in  general  joint since  the  i n v e r s e dynamics appear i n g e n e r a l  a sum o f p o l y n o m i a l s this  work  use  over t h e whole of a m a n i p u l a t o r ' s  i n Cartesian  coordinates,  b u t more d i f f i c u l t  learning  of  difficult  to learn,  Cartesian system,  coordinates  also  or i t s equivalent  kinematics.  using  the manipulator  coordinates  f o r use i n e r r o r c o r r e c t i n g  or i t s equivalent  translate directly The  d i r e c t p o s i t i o n kinematics  be w e l l r e p r e s e n t e d  of  the  using  manipulator's  manipulators  consist  The  manipulator  reach.  i n general  kinematics  a  position  position in feedback.  accurate  vision  Cartesian  The  as  vision  inaccuracies  i n t o path t r a c k i n g e r r o r s .  can  terms.  must be v e r y  the  Path s p e c i f i c a t i o n i n  the a n a l y t i c a l d i r e c t  to  self-  contain  n e c e s s i t a t e s t h e use of  kinematics,  system  observe  i t requires  i n v e r s e dynamics which  inverse  path  w h i c h i s more c o n v e n i e n t  t o implement as  the Cartesian  by  reach. In  we h a v e i m p l e m e n t e d s e l f - l e a r n e d c o n t r o l u s i n g  specification to  t o be w e l l r e p r e s e n t e d  of  direct  o f t h e two l i n k  sums o f p o l y n o m i a l s  over t h e whole  T h i s a p p e a r s t o be  since their  possible  analytical direct  sums a n d p r o d u c t s o f  position  manipulator  kinematics  sine of  were s e l f - l e a r n e d and used i n p l a c e  of  position  and  the  for  cosine  two  link  the  vision  system i n c l o s e d loop c o n t r o l . T a b l e 3.22 s u m m a r i z e s t h e p a t h e r r o r , a v e r a g e d o v e r t h e s i x standard  paths,  schemes.  I t c a n be s e e n t h a t p e r f o r m a n c e u s i n g  Cartesian using control  that  r e s u l t s when u s i n g  the  various  the s e l f - l e a r n e d  i n v e r s e dynamics i s q u i t e comparable t o t h a t  the  analytical Cartesian  using  i n v e r s e dynamics.  the s e l f - l e a r n e d Cartesian  168  control  inverse  achieved  Closed dynamics  loop in  conjunction is  a  very  position  with  viable  an  control  kinematics  performance  is  self-learned  direct  to  technique  significantly position  the  are  used  position  applied  and  the the  achieve  b e t t e r performance  direct  position  achieved  using  kinematics higher  system, When  the  i n s t e a d of  reflecting  We  i n the  accuracy  of  resulting only  self-learned  more  order,  representations.  169  vision  that stages  the  an of  the  of  the  i t serves  averaging  or  self-learning direct  performance.  accurate, sum  in  self-learned  control  direct system,  inaccuracies  requires that  be  i t s equivalent,  Nevertheless  found final  or  the  the  kinematics.  technique.  improved  kinematics  vision  technique.  degraded,  demonstrate  smoothing  accurate  To  self-learned as  could  be  polynomials  XERMS  VXERMS  6.48E-4  2.10E-3  8.06E-2  Standard  3.52E-3  5.22E-3  9.10E-2  Pre-Learned Dynamics  4.23E-3  1.01E-2  1.18E-1  Self-Learned Dynamics  3.37E-2  3.92E-2  2.15E-1  Pre-Learned Cartesian Inverse Dynamics & Pre-Learned D i r e c t Position Kinematics  7.37E-2  8.60E-2  2.89E-1  Self-Learned Cartesian Inverse Dynamics & S e l f - L e a r n e d D i r e c t Position Kinematics  2.96E-2  4.11E-2  5.67E-1  Self-Learned Cartesian Inverse Dynamics & Smoothed SelfLearned Direct Position Kinematics  Table  3.22  Summary  AXERMS  of path  CONTROL  errors  170  using  SCHEME  Analytical  Control  Cartesian  Cartesian  various  Inverse  Inverse  control  shemes  4 CONTRIBUTIONS OF THIS  This related  thesis  of a r o b o t be  done  without  learning  of  corresponding  I t h a s been shown t h a t t h i s  recourse  to analysis  the manipulator.  based  on  Interference have  Minimization  various  numbers  estimates  The viability  of  various  called  A new v a r i a n t  Pointwise  h a v e been  the  other  Interference  described.  c o n c e p t o f s e l f - l e a r n i n g h a s been i n t r o d u c e d demonstrated using  of  Interference  and compared w i t h  A p p l i c a t i o n s where P o i n t w i s e  i sappropriate  algorithms  o f i n p u t v a r i a b l e s . A method h a s  been i n t r o d u c e d  learning algorithms. Minimization  the principal  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  Minimization has  I t has  Identification  and these r e l a t e d l e a r n i n g  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 .  Interference  and  self-learning.  Method and L e a r n i n g  been d o c u m e n t e d f o r r e d u c i n g  It  torques  u s e d i n t h i s t h e s i s . The c o n v e r g e n c e r a t e s o f  Minimization  having  through  h a s been s t u d i e d .  b e e n c o m p a r e d f o r sum o f p o l y n o m i a l s  orders,  when  applied  s i m p l i f i c a t i o n s of Interference M i n i m i z a t i o n ,  learning algorithm  learning can  i s possible  of  f a m i l y of l e a r n i n g algorithms  kinematics  t h e dynamics and  m o t i o n ; h e n c e t h e name  been shown t h a t t h e G r a d i e n t are  of  This  observation  manipulator  and  c a n be u s e d t o l e a r n t h 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 t h e dynamics and  manipulator.  kinematics  A  h a s 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  learning algorithms  multi-variate  THESIS  a simulated  two l i n k  and  i t s  manipulator.  h a s b e e n 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 t h e  inverse  dynamics,  the inverse  kinematics  171  (which together  form  the  Cartesian  kinematics  inverse  of  the  dynamics)  two  link  order  sums  and  the  manipulator  direct  can  position  be  adequately  th represented over  by 4  a l l of  thereof  sufficient  learning  of  the  acceleration  given  The  Cartesian  using  point  accelerations manipulator position  as and  vision  the  accelerations Cartesian  direct  A  simple  range  during  inverse  dynamics  of a p p l i e d  direct  as  by  utilizing  torques  and and  of  The  the direct points  positions using  demonstrated to  be  and a  that  constrained  velocities,  and  o f s e l f - l e a r n i n g when  the  a r e n o t known shown t h a t  172  fc  dynamics.  observed  positions,  during  4 ^  good  training  joint  motion  kinematics  as  system.  end p o i n t  I t has been  a  dynamics  has been  stages  only  using  velocities  positions  s e l f - l e a r n i n g c a n be r e p l a c e d position  inverse  technique  the early  with  by u t i l i z i n g  of Cartesian  resolved  specifications  t o be a l m o s t  measured  Self-  Performance  dynamics  a vision  Cartesian  manipulator  manipulator.  during  point  of  path  done  portion  permits  c a n be l e a r n e d  the  c a n be l e a r n e d  end  a desired  by  a  dynamics  positions,  using  n-tuples  system.  permits within  observed  of  corresponding  Cartesian  or  workspace.  Cartesian  dynamics  imposed  kinematics  consisting  found  c a n be  a useful  c o n s i s t i n g of n-tuples  end  reach  using  inverse  has been  inverse  This  coordinates.  the a n a l y t i c a l  points  resulting  the  in Cartesian  Cartesian  training  inverse  t o be a p p l i e d  representation  performance  manipulator's  Cartesian  self-learned  order  link  t o be c o n s i d e r e d  control  conveniently the  t h e two  of polynomials.  with  well  enough  the v i s i o n the f i n a l  subsequent  to control  system  used  self-learned use  of  the  manipulator. While simulated would  self-learned two  lead  approach  that  freedom  is  convenient  following  manipulator,  evidence  has been  path  f o r the tasks suggestions chapter,  adaptation  implementation  of  modification  inspiration  that  self-learned  for future  of for  5)  an  t h e r e q u i r e d sum of  for this  the Klett thesis)  general  having  more  degrees  173  that  are presented such  control -tool  mass),  of polynomials  Cerebellar  to increase  system  out.  covering  unknown  that  i s a  research  self-learned  for a  presented  in a coordinate  t o be c a r r i e d  chapter  demonstrated  control  to manipulators  specification  investigation  automatic  and  been  c a n be a p p l i e d  using  Detailed  further  has only  one t o b e l i e v e  of  the  link  control  Model  (in  topics  as  (especially hardware estimators, (the  i t s plausibility.  major  5 S U G G E S T I O N S FOR F U T U R E  The  characterization  formulation link  and demonstration  manipulator  principal  5.1  are  the basis  CONTROL From  would  certain  prove  a  manipulators  no  having  manipulator  having  step  would  real  manipulator of  further. the  rate  feedback in  of to  more  terms  of  work  scheme  for  complex  of freedom. test  self-learning  progresses.  as may  have  been  of  to apply  made  that  many  minimizing  to  real  logical  be  to a  studied  had an e f f e c t by  excursions,  tracking  error,  It i s also  apparent  that  on  changing  feedback  path  174  a  control  experiment  bounds  would  freedom.  parameters  of  The  self-learned  One m i g h t  out  to  link  believe  that  control  of  two  We  freedom.  2 degrees  control  industrial  vehicle.  of s e l f - l e a r n i n g need  minimizing  while  that  The  s e l f - l e a r n i n g a s t h e optimum  instability,  just  self-learned  self-learned  than  of self-learning. during  perhaps  the  CONTROL  that  6 degrees  the aspects  gains  is  MANIPULATOR  a preliminary  I t was a p p a r e n t  algorithms  to this  OF S E L F - L E A R N E D  be t o a t t e m p t with  o f t h e two  research.  assumptions  up  algorithms and  control  however,  related  up t o 6 d e g r e e s  extension  next  Many  thesis,  control  was m e r e l y  the  learning  i t was h o p e d  viable  simplifying  prohibit  these  OF A MORE COMPLEX outset  of l e a r n i n g  of s e l f - l e a r n e d  proposals  for future  the  manipulator  family  of t h i s  FURTHER I N V E S T I G A T I O N  5.1.1  that  utilizing  contribution  important form  of a  RESEARCH  changes  gain i e . as  the choice of  training  paths,  forcefulness learning  or  bounds  discussed  in  as  of  example  the  out  bounds  where  using  this  discussed  in  It  achievable  algorithms polynomials.  on  The in  learning  time,  control  action,  etc.  will  manipulator.  5.1.2  ADAPTATION initial the  terms  TO  learning  of  inverse  dynamics.  of  a  A  real  but  be  required  was  found  may  to  upon  etc. to  For  limit  during not  been  polynomials learning  other the  than  numerous  capability, of  of  estimation  use  evaluating  out  techniques  of  functions  apparent  of  such  occur  mean  adequacy  here  situations  i t has  sum  computing and  be  averaging  possible  when  used  risk  the  and  warranted  manipulator,  that  the  minimize  technique;  using  apply  is  excursions  order  on  simplistic  to  there  and  constraining  technique  here  improve  given  also  to  very  order  bounds  can  become more  inverse  a  apparent  3.9.1  the  excursions,  well  performance,  intent  in  constraining  of  TOOL  was  used  estimation  costs  real  out  may  possibilities  The  quite  with  It  based  damage  technique  effect  3.9.1,  constraining  bounds  is also  section  representation.  of  might  excursions  otherwise.  accuracy  out  an  Investigation  techniques  unacceptable  self-learning  simulations  the  iterations,  have  section  proposed.  better  that  learning  action  in  of  constraining  large  proven  were  maximum  excursions  of  these  parameters  development  quantities  as  As  techniques  changing  of  constraining  used  improved into  of  rate.  technique  frequency  self-  constraining  application  to  a  MASS of  this  thesis  dynamics  vision  was  to  demonstrate  or  the  more d i f f i c u l t  system  was  to  175  be  used  for  self-  Cartesian feedback  purposes.  It  applied  in  direct or  soon other  position  free  high  i t for  area  adaptation  that  of  that and  the  m  The  mass  2  =  link  link  +  lengths,  tool  the a  required the  two  mass tool  learn  replace of  the  achieve  to  link  the  the  of  vision  fine  tool  the  path  etc.  with  were  Cartesian  other  One  mass  inverse  input  be  of is  can  the  variable  in  in  final  parameters  One  can  the  imagine link  holding,  constants  dynamics.  where  mass.  manipulator to  system,  self-learning is  manipulator  considered  the  tracking.  (  along  be  estimate  workspace  manipulator.  consists that  to  the  a p p l i c a t i o n of  according the  to  areas  to  self-learning could  tool  m  another  s  thus  specific  invites  .the  masses  forming m  link  m  is  that  example  and  in  link  of  for  use  again  second  the  areas,  control  Consider  apparent  kinematics  resolution  Andther  became  in  5  '  1  )  such  as  functions  also  think  Cartesian  of  inverse  dynamics, r Given the  cart.inv.dyn(x,x,x,m  that  m t  o  ^  a  n  inverse  manipulator Once  good  mass  of  and  thus tool  might  attach  d  while  control  the  for  to  (5.2) should  for  a  range  initially  tool  of  the  be  able of  i s achieved,  could  To the  tool the  and  of  then  correspondingly  Cartesian  adjust mass  one  inverse  the  Cartesian  the  tool  176  masses.  inverse begin the  inverse one  A  variable,  mass  to  the  dynamics. to  vary  variable  dynamics  i n use,  self-learn  input  corresponding Cartesian  to  tool  f i x the  s e l f - l e a r n i n g the  attached  masses.  be  ^) one  dynamics  a  self-learn  appropriate  t o o  is specified,  technique  tool'  of  o  Cartesian  good m  =  for  a  dynamics need  only  the m t  o  o  i  range to  be  know  m  tool  a  n  learned  d  a  PP^  ^  v  Cartesian  A  logical  determined torques.  s  a  next  motion  manipulator  could  unknown  i s attached,  as  given  in  relationship  A  that  look  representable sum  of  be  were  self-  the  o  i  on  desirable of  two  through  for  m  the  when  a  link  a  the It  whole  portion  that  was  this of  the  tool  of  One  could dynamics  a  functional  dynamics,  of  the  chosen If  function  the  likely,  representable.  of  (5.3)  that  is  of  manipulator  inverse  yielding  tool  be  applied  control  (3.66).  the  tool  and  can  capability.  the  into  mass  observations  then  appropriately  (3.51)  call  motion  based  torques  tool  will  probably  manipulator's however,  that  manipulator's  so  that  the  so,  then  the  reach i t can  reach,  tool  using be  such  Cartesian  not  so as  inverse  dynamics  could  self-learned. The  stop  process  here.  analogous applied force and  over  over  workspace  dynamics  o  dynamics  suggests  polynomials.  represented the  t  (5.1)  will  whether  m  very  solve  we  final  tool.dyn(7,a,a,a)  =  preliminary  be a  tool  a  inverse  and  the  manipulator  adjusted  equation  relationships  ask  applied  equations  substitute  m  and be  the  to  discern  manipulator  Consider  is  observing  could  of  dynamics.  step  by  one  mass  input. variable  n  inverse  simply If  a  Clearly, to  force  force one  hence  of  for  needs one  generalization s e l f - l e a r n i n g of  feedback. the  to  unknown  estimate  must  learn  One  of  the  merely  applied  self-learning tool  several  177  dynamics  substitutes  t o o l mass. With  i t s d i r e c t i o n as  need  well  functional  as  is  an  not quite  unknown  an  applied  magnitude  relationships.  Conceptually, A  wealth  apparent.  applications  relationships  required  to when  of  polynomials  using  sum  manipulator  fields,  anywhere  multi-variate  t o be d e m o n s t r a t e d .  input  may  functional  of the s e l f - l e a r n i n g i n  be  of  variables..  only  portions  but  calculations In other  be r e p r e s e n t a b l e of t h e i r  with.the  I n some  applicable  o f t h e number  to apply  the  reason  OF  POLYNOMIALS  as  domains  learning  cases sums making  algorithms  ESTIMATORS  for investigating representations  computational  Interference  Gradient  polynomials f  SUM  of polynomials  resulting  a  narrow  difficult  principal  algorithms,  that  over  I M P L E M E N T A T I O N OF  and  many  of  are  here.  A  In  with  self-learning  i n other  theoretically  because  similar.  the f i e l d  exist  remains  may  be for  The v i a b i l i t y  t o be e s t i m a t e d  self-learning given  well  however,  dealing  functions  from  non-linear,  implement  the  the  may  self-learning  difficult  are  i s required.  applications,  cases  5.2  of  would  applications  examples  learning  these  the technique  of p o s s i b l e  These  control; that  though,  i s the  structure.  Minimization,  Method  self-learned  a r e a l l based  The  regularity three  Learning on  Identification  the  same  of  (5.4)  control the  appropriate  sum  estimator,  T  coefficients  of  learning  = w p  i s  control  application  i t i s the evaluation  time  critical  can  take  task.  place  f o r the computing  Learning  off-line  capability  178  of the of  the  utilizing available  estimate weighting  a  method  and the  rate  of  learning  converges  f o r example,  much f a s t e r  complex, to  desired;  and  the G r a d i e n t Method but  thus time consuming weight  the r e g u l a r i t y  applicability warranted  than  of t h e sum  to  problems of v a r i o u s s i z e s ,  into  been  done  hardware  and  adjustment  the  implementation  for  implementation  stochastic  computing  of  sum  formula.  of  hardware. using  more  and  Due its is  polynomials  P r e l i m i n a r y work  digital  hardware.  computing  More  work  is  approaches.  D I G I T A L COMPUTER IMPLEMENTATION In  factor sum  uses a  investigation  r e q u i r e d t o demonstrate the v i a b i l i t y of these 5.2.1  Minimization  of p o l y n o m i a l s e s t i m a t o r  e s t i m a t o r s using s p e c i a l l y developed has  Interference  a  digital  computer i m p l e m e n t a t i o n ,  i s t h e number o f m u l t i p l i c a t i o n s  of  polynomials.  polynomials  does  Fortunately,  not  the r a t e  limiting  required to evaluate  e v a l u a t i o n of  the  r e q u i r e a s many m u l t i p l i c a t i o n s  the  sum  of  as  one  m i g h t assume. Recall  that,  for  a  system  of o r d e r  s  having  v  input  v a r i a b l e s , the p o l y n o m i a l terms are the s e t , v {p,(z)} = { n z . i } * i=0  (5.5)  e  where, v Z£ e. =*= s ,. i-0 z The Each  n  e, an ii nn tt ee gg ee rr e.  (5.6)  1  = 1  (5.7)  i n p u t v a r i a b l e s and  the constant  1 form the  of the remaining h i g h e r o r d e r terms can  product  o f two  lower order terms.  179  The  total  be  first  v+1  formed  terms. as  the  number o f t e r m s i s ,  (s+v)! m =  (5.8) v! s!  Table  5.1  shows  t h e number o f t e r m s f o r  systems  of  various  o r d e r s , s, a n d h a v i n g 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 . The number i s m -  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 t h e p o l y n o m i a l (v+1).  In terms  a  manipulator  will  typically  functions.  In  manipulator variables  control application be u s e d i n  modelling  with will  r  the  t h e same  estimation  of  degrees of freedom, corresponding  t h e number to  an  necessary  one  f o r each degree of freedom.  pass  several  requiring  m-v-1  of  of  input  There w i l l F o r such  multiplications  t o form t h e p o l y n o m i a l terms and r passes r e q u i r i n g  multiplications  and a d d i t i o n s each a r e n e c e s s a r y  a  acceleration,  f u n c t i o n s forming the C a r t e s i a n i n v e r s e dynamics.  manipulator,  polynomial  the C a r t e s i a n i n v e r s e dynamics  be v = 3 r ,  v e l o c i t y and p o s i t i o n r  terms  to  be a is m  accumulate  t h e w e i g h t e d sum o f t h e s e t e r m s t o c o m p l e t e t h e e s t i m a t e s o f t h e r  functions.  forming  For e s t i m a t e s of order s f o r each of t h e f u n c t i o n s  the  multiplications  Cartesian  inverse  dynamics,  the  number  of  required i s thus, (r+1)(s+3r)!  cid(r,s) =  - (3r+1)  (5.9)  s!(3r)! In  modelling the d i r e c t p o s i t i o n  having r degrees of freedom, be v = r , c o r r e s p o n d i n g There w i l l For  kinematics of a manipulator  t h e number o f i n p u t v a r i a b l e s  t o the p o s i t i o n  f o r each degree of freedom.  be r f u n c t i o n s f o r m i n g t h e d i r e c t p o s i t i o n  e s t i m a t e s of order s f o r each of the f u n c t i o n s 180  will  kinematics. forming  the  direct  position  kinematics,  t h e number  of m u l t i p l i c a t i o n s  is  thus, (r+1)(s+r)! dpk(r,s)  =  -  (r+1)  (5.10)  s! r !  System 1 N u m b e r  1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18  0 f V a r i a b 1 e s  Table and  5.1  Number  various  Table to  2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19  5.2  evaluate  estimators  required  of  4  3 6 10 15 21 28 36 45 55 66 78 91 105 120 1 36 153 171 190  4 10 20 35 56 84 120 165 220 286 364 455 560 680 816 969 1 140 1330  5 15 35 70 126 210 330 495 715 1001 1365 1820 2380 3060 3876 4845 5985 7315  to  table  evaluate  f o r sytems  5  6 7 28 84 210 462 924 1716 3003 5005 8008 12376 18564 27132 38760 54264 74613 100947 134596  6 21 56 126 252 462 792 1287 2002 3003 4368 6188 8568 1 1628 15504 20349 26334 33649  of v a r i o u s  orders,  of m u l t i p l i c a t i o n s  of the C a r t e s i a n  orders,  s,  numbers  summarizes the  .  s,  variables, v  t h e number  various 5.3  m,  input  summarizes  of various  to  3  the estimates  corresponding Likewise,  2  of terms,  numbers  Order  dynamics f o r  and having  input  of degrees  of  t h e number  estimates  181  inverse  of  of the  required  variables  freedom,  r.  multiplications direct  position  kinematics.  Sytem  D e  F r g o e r f e e d e o m s  1 2 3 4 5 6  Table  5.2  orders  1  2  3  4  5  6  4 14 30 52 80 11 4  16 77 210 442 800 1311  36 245 870 2262 4880 9291  66 623 2850 9087 23240 51 186  108 1379 7998 30927 93008 235524  1 64 2765 20010 92807 325568 942153  Multiplications  of the C a r t e s i a n  various  degrees  Order  of  required  i n v e r s e dynamics  F r o e f e d o m  1 2 3 4 5 6  5  6  4 9 16 25 36 49  6 18 40 75 126 196  8 30 80 175 336 588  10 45 140 350 756 1470  12 63 224 630 1512 3234  14 84 336 1050 2772 6468  of the d i r e c t  such  as  4  multiplications corresponds  of  required  to  position  two  kinematics  model  of a  link  2 degree  estimates  of  various  for manipulators  usee  of  manipulator  a r e r e q u i r e d . Assuming 16  for  having  freedom  order  the  Order 4  Multiplications  a  having  3  orders  For  for manipulators  2  5.3  degrees  various  1  Figure  various  of  freedom  System  D e g r e e s  for estimates  per  that  freedom  studied t c a i c ^ *  multiplication.  182  manipulator  If  here, 0  1  s  e  c  »  623 this  self-learned  control 51186  can  be  multiplications  this  corresponds  rate,  but  i f  Evaluation  of  fewer  a  195  only the  basic  direct  purpose  an  fetching  of  operands  portions  of  operation; general  i f only  purpose  would  be  which  is a  task,  is  algorithm  low  point  less  time  assumed  is easier  a  utilizing 5.1  digital  of  be  be  general  be  the  but  performed supervising  shows a  block  hardware.  would  two  modes  of  polynomial  terms  i s generated.  K-G  supervising  computer  constant  1  and  memories.  It  i s assumed  to  the  within  operand the  the  operation.  v  input  memories  sequence  digital  places  that and  used.  the  for  to  high  required,  computer.  one's  standard system  Learning,  chosen  sum In  of  the  of  polynomials  the  complex learning  To  proposed  first  begin  mode,  this  terms,  digital estimator  the  set  operation,  consisting  into  p o i n t e r memories  183  a  A l l speed  dedicated  the  two  s u p e r v i s i n g computer  and  perform  memory.  very  a  computer.  variables,  controller  are  algorithmically  using  initial  the  and  This  purpose  diagram  computing have  The  were  requires  implementation  controller  designed  more  necessary.  estimator  results,  operation  could a  of  high  achieve.  computer  sequence  a  technology,  estimates to  sec,  1  quite  are  polynomials  a  t ^ i ^ O ' ^  operations  thus  of  manipulator,  current  and  critical,  to  with  kinematics  speed  by  For  position  would  computer  freedom  multiplication,  storage  system  of  required.  per  sum  and  supervised  Figure  be  accumulator,  this  degree  achievable  fixed  multiplier,  6  nsec  components  general  a  would  multiplications  The of  to  to  nevertheless  especially  far  extended  as  accumulator.  well The  of  of the the  operand  has  access  registers  higher  order  address  2  SEQUENCE CONTROLLER address 1  \  /  \.  POINTER MEMORY 2  POINTER MEMORY 1  \  /  \  \  /  \  /  \ result  MULTIPLIER  /  Figure a  /  \  \  /  5.1  operand  1  operand  2  Block  result  \ /  S E T  \  diagram  sum o f p o l y n o m i a l s  \  /  ACCUMULATOR p 0 L Y N 0 M I A L  estimator  184  /  /  P 0 L Y N 0 M I A L  W E I G H T  W E I G H T  S E T  S E T  S E T  1  r  OPERAND MEMORY 2  OPERAND MEMORY 1  of d i g i t a l  /  computer  implementation  of  terms of  i n the s e t of polynomials  lower  the  order  two  lower  terms  operand  order  memories.  memories.  The s p e c i a l i s fixed  specific  number  generated  indexes  through  each  lower time. both  are  the second  now  polynomial of  from  completion different evaluated memory  These  terms  term  operands  weight  i s assumed  read  from  those  the  s,  pointer  and having  addresses  both  of  a  c a n be in  the  controller  easily terms  of  i n the pointer  sequence  generated  i s duplicated the  required  t o be w r i t t e n back  weighting products  pass. set  using  each  terms.  estimate  from  i s  The two o p e r a n d  directly  operand  read  memory operand  coefficients a r e summed  by  into  Several  each  this  such  time Note  2  185  of  the  consist  f o r the function  to  i n the accumulator  and  passes  a r e made  until  estimates  that  t h e number  approach i s ,  sequence  1 consist memory  formed  memories  the  by t h e s u p e r v i s i n g c o m p u t e r  for a l l r functions.  required  pairs  and p l a c e d  with  in  c a n be f e t c h e d a t t h e same  of operation,  the accumulator of each  products  simultaneously.  while  Their  the  so that  sequentially,  Operands  estimated.  read  memories  sum o f p o l y n o m i a l  the corresponding  be  operation,  result  mode  addressed  controller.  order,  v.  by  stored  The s e t of p o l y n o m i a l  order  memories  a weighted  of given  p o i n t e r memories  polynomial  The h i g h e r  In as  the  of the operand  operand  performed  variables,  During  addresses.  order  i s  by t h e s u p e r v i s i n g c o m p u t e r  memories.  as  of a p p r o p r i a t e  p a t t e r n of addresses  of input  once  sequential  terms  in turn  t o the s e t of polynomials  Selection  f o r a system  pointer  in  and appended  polynomial  memories  a r e formed  at  the  using  have  a  been  o f words o f  (r+4)(s+2r)! mem(r,s)=  (5.11) s!r!  A for  key f a c t o r  In  approximately learned were  7 decimal  estimates  about  2  or  simulations  of  of t h i s  approach  required i n the  self-learned  control,  d i g i t s o f p r e c i s i o n were u s e d . Our  of the C a r t e s i a n i n v e r s e  dynamics,  d i g i t s and y e t  were  Input v a r i a b l e s a r e normalized  self-  however,  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 3 decimal  control purposes.  functions  adequate between  for  -1 a n d  a n d 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 o u l d a l s o be n o r m a l i z e d .  would  thus  precision  appear t h a t f i x e d p o i n t b i n a r y  control  purposes. at  necessary,  Such f i x e d p o i n t o p e r a t i o n s  the rate required using current  higher p r e c i s i o n  digital  operations  o f p e r h a p s 8 t o 10 b i n a r y d i g i t s w o u l d be  achievable  be  our  only accurate  to  for  the v i a b i l i t y  very h i g h speed o p e r a t i o n i s t h e p r e c i s i o n  computations.  1,  i n determining  It  with  a  sufficient should  be  technology.  If  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 f o u n d i t w o u l d be much more d i f f i c u l t  sum o f p o l y n o m i a l s  to  t o implement t h e  estimator.  5.2.2 STOCHASTIC COMPUTER IMPLEMENTATION It  may a l s o be p o s s i b l e t o i m p l e m e n t a sum o f  estimator u s i n g s t o c h a s t i c computing techniques. computer  [19,48] i s d i s t i n c t  computer  as  occurrence  numbers of  a  1  For and  B,  example, which  are  by  i n the b i t sequences  permitting multiplication  The s t o c h a s t i c  form e i t h e r t h e analog  are represented  the on  between  0 andM,  186  or  digital  probability the  t o be p e r f o r m e d by a s i n g l e  consider a situation  polynomials  data  of  lines  gate.  i n w h i c h t h e numbers  are  represented  as  A  the  probabilities  of  the  occurence  of  a  1 on  their  respective  data  lines,  If  Prob(A=1)  =  A/M  (5.12)  Prob(B=1)  =  B/M  (5.13)  these  will  be  output  signals  a  1 whenever  being  a  the  fed  into  both  an  inputs  AND  gate  are  1's.  then The  its  output,  C,  p r o b a b i l i t y of  the  1 is,  Prob(C=1) If  are  =  Prob(A=1  probabilities  on  and  B=1)  the  (5.14)  input  lines  A  and  B  are  independent  then, Prob(C=1) and  the  number  product C This  =  of  A  is  a  represented  and  = AB/M  is using also  a  be  logic  an  N  sequence  i s the  normalized  EXCLUSIVE  input a  with  Normalized  simple  circuitry.  sequence  The  successive  whose  signed  stochastic  computer  sequence  generator  is  by  representation number  gate  a  b i t binary random  This  of  [19].  for  pseudorandom  form  OR  similarly  signals  bipolar permits  [19].  b i t binary  A  which  circuitry  generator.  Bernoulli input  N  possible  performed  using  an  line  representation.  single  be  formed  of  also  random  comparing with  form  Adequately  suitable  output  (5.16)  multiplication  can  the  B,  unipolar  can  on  (5.15)  2  representation  summation  Prob(A=1)-Prob(B=1)  done of  obtained  the from  r e s u l t s of  p r o b a b i l i t y of  a  repeatedly  input a  and  signal  pseudorandom  comparison  1 corresponds  form  a  to  the  can  also  signal. We  have  shown  that  adequately  187  random  input  signals  be  obtained  jitter  by r a n d o m i z a t i o n  of pulse-rate-encoded  signals  using  stages [23],  Simple  hardware  multiplication technique.  and  Its  Consider of  summed  (more  A  pairs  of  =  a  make  stochastic  parallel  processing  i s  attractive  for  VLSI  drawback,  however,  i s  that  may  given  be v e r y  slow.  in figure  5.2.  of inputs and these  correctly  of a  Prob(G=l)  makes  principal  example  operation  a potential  and summations  two  probabilities probability  summation  the  formed  parallel  simplicity  implementation. multiplications  and  1  at  averaged).  products Assuming  a l l inputs  1 at the output  Products  are  are  are then  that  the  independent,  the  Gi s ,  [Prob(A=1)-Prob(B=1)  +  Prob(D=1)-Prob(E=1)]/2 (5.17)  This  structure  polynomials it  N  samples  Since  halve  =  use of the r e s u l t  this  probability  occur  of the p r o b a b i l i t y  a  at the  of  output the  F o r example, i f  and 0 of these of the output  i s a Bernoulli  sequence,  samples being  a  1  are i s ,  the standard  deviation  of the p r o b a b i l i t y i s ,  [Prob(G=l)(l-Prob(G=D)/N]  one must  Thus  sum  by e s t i m a t i n g  at the output.  of the output  the standard  probability output.  implement  (5.18)  estimate  SDEV  to  = O/N  the output  this  extended  T o make  to estimate  a r e taken  an e s t i m a t e MEAN  To  be  p r o p o r t i o n o f 1's t h a t  1's,  in  estimator.  i s necessary  mean  could  while  deviation  quadruple stochastic  i n one's  t h e number computing  188  (5.19)  1 / / 2  estimate  of t h e output  of observations of the may  be  comparatively  fast  f o r low  variables, are  computations  i t becomes v e r y  involving  many  slow i f high accuracy  input  computations  required. As  of  accuracy  discussed  estimates  great.  Thus  polynomials fixed  i n s e c t i o n 5.2.1, i t a p p e a r s t h a t t h e a c c u r a c y  required stochastic  estimator  point  to achieve  adequate c o n t r o l i s not  computer implementation of  may be f e a s i b l e .  arithmetic  in a digital  a  very  sum  of  N o t e t h a t , a s when  using  computer,  range  narrow  r e s t r i c t i o n s a r e i m p o s e d 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 stochastic  computer.  Further  determine the f e a s i b i l i t y  is  required  to  of such an a p p r o a c h .  >  B  investigation  -CL  INPUTS ARE INDEPENDENT BERNOULLI  SEQUENCES  y  )  PSEUDORANDOM  Figure  5.2  S t o c h a s t i c computer c i r c u i t  o f two p a i r s o f i n p u t  variables  189  f o r summing t h e p r o d u c t !  5.3 MODIFICATION OF THE KLETT CEREBELLAR MODEL  The  starting  Model. During  p o i n t f o r t h i s work was t h e K l e t t  the course  Cerebellar  of t h i s research, s e v e r a l i n s i g h t s  o c c u r r e d a b o u t ways o f m o d i f y i n g  have  t h e C e r e b e l l a r M o d e l t o make i t  more p l a u s i b l e a s a m o d e l o f t h e mammalian  cerebellum.  5.3.1 THE KLETT CEREBELLAR MODEL Klett polynomials shows  [ 2 6 ] models  t h e mammalian c e r e b e l l u m a s a  estimator using orthogonal  polynomials.  a block diagram of t h e c e r e b e l l a r  system.  sum o f  Figure  5.3  The C e r e b e l l a r  Model i n c l u d e s o n l y those 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 t h e p r i n c i p a l  pathways.  PARALLEL FIBERS  GRANULE CELLS  GOLGI CELLS  75+  CEREBELLAR CORTEX  BASKET t STELLATE CELLS  \  -V  SUBCORTICAL NUCLEAR CELLS  KEY -J- EXCITATORY SYNAPSE MOSS; FIBERS  F i g u r e 5.3  —  INHIBITORY SYNAPSE  TO MOTOR CENTERS VIA VARIOUS ROUTES  Block diagram of c e r e b e l l a r  190  system  CLIMBING FIBERS  A 5.4.  block  d i a g r a m of t h e C e r e b e l l a r M o d e l i s shown i n f i g u r e  Purkinje  granule  cells  c e l l s a r e e x c i t e d by  the p a r a l l e l  and  b a s k e t and  i n h i b i t e d by  the  w h i c h a r e a l s o e x c i t e d by p a r a l l e l be  equivalent  the  i f Purkinje c e l l s could  parallel  be  P u r k i n j e c e l l s and  stellate  are  cells  f = I w^q^  net  need f o r  the cells  e f f e c t would inhibited  by  basket  and  t h e i r associated basket  and  t h u s m o d e l l e d as  e i t h e r p o s i t i v e or n e g a t i v e  The  stellate  e x c i t e d or  f i b e r s thus e l i m i n a t i n g the  stellate cells.  having  fibers.  f i b e r s of  w e i g h t e d summation  weighting  points  coefficients,  = wq  (5.20)  T  k  It  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  corrective take  climbing  fiber activity  that  coefficients  by  l e a r n i n g i s assumed  to  place, Aw  Note  = uAfq  that  available  such  (5.21) l e a r n i n g i s performed based  information  i n each P u r k i n j e  on  only  cell.  CLIMBING  Figure  5.4  Block  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 o d e l  191  locally  Granule cells  cells  are  as  themselves -  Golgi q  Mossy  =  the  by  matrix an  This  In  Golgi  [/ S  pp  Klett  cell  [26].  G  - Golgi  are  chosen  cell  The  variables  i s summed  and  the  Golgi  negtive  The  T  T  as  results  6S]~  1  G  6S]  /  is  Granule  cell  network,  is  thus,  synaptic weights  /  be  in  -  2  improved 2.  I  Such  (5.25)  Model, are  these  special  assumed,  that  performed  not  Parallel  by  possible  analogous  [19,48].  192  neurons to  to  and  and  perform  variables  fiber  learned.  multiplication  pulse-rate-encoded are  Q  as,  of  techniques  results  matrix  previously in chapter  summation  that  the  i f ,  chosen  evidence can  This  the  represented  (5.24)  be 1  shown  certainly  gates  result are  2  Cerebellar  here  with  cells  feedback  a p p r o p r i a t e l y then  is  computing  a  activity  It  logic  activity  Granule  (5.23)  synaptic weights  There required  as  the  product  activity.  orthogonalization matrix.  [J pp S  =  the  fiber  fiber  requires that G  Cell  fiber  i s modelled  performance  =  This  of  The  = Qp  orthogonalization Q  product  activity.  parallel  Parallel 1  learning  Golgi  fibers.  (5.22)  (I+G)" p  becomes  the  Mossy  - Gq  Parallel  the  the  activity.  fiber  network  state  q  fiber  by  by  forming  inhibitory  excited  p  as  Parallel  cell =  Steady  If  by  negative,  output  excited  modelled  represented the  are  summation  neural  networks  multiplication with  neurons  simple  using  or  digital  stochastic  5.3.2  LEARNED A  it  ORTHOGONALIZATION  principal criticism  assumes  Golgi  the  cell  special  network.  implausible  as  information  to  To  Without  equivalent Gradient  to  the  is a  rates.  accurate  of  Golgi  optimal  have  the  in  a  cerebellum.  that  is locally  5.5a  shows a  The  of  found  learning  in  in  can  the  take  becomes 2.  The  terms  of  learning  training.  i t is possible could  in of  enough  learning  to  chapter  capable  that  that  Model  algorithm  given  available  rather  amount  Cerebellar  is  manner  cell  Model  have  learning  that  that  weights.  permit  i t  is  Granule  enormous  outlined  estimates,  found Q  to the  Method  the  we  neccesary  less  matrix  an  these  Nevertheless,  we  orthogonalization  information  not  in  Model  Cerebellar  argument  polynomials  Furthermore,  in  specify  Gradient  convergence  performed  the  require  this  Cerebellar  weights  makes  would  is  Klett  orthogonalization,  Method  sum  synaptic  genetically  orthogonalization  the  This  it  counter  place.  of  to  learn  the  conceivably be  done  Granule  be  using  cells  and  cells. Figure  cell  network  occur.  showing  Figure  Parallel  q  fiber  =  a  lumped  that  (b.p.  with  those  an  as  much  the as  the  Granule  where  same  cell  With  cells  Golgi could  but  this  with  model  the  by,  - L g..q.) J  -  amplification  schematic  possible.  i s given  (5.26)  1  individual Granule  Golgi  of  locations  shows  activity  j*i Note  those  5.5b  amplifications  schematic  that  cell  i s assumed  i t i s being  193  inhibited  to  not  by.  synapse  Figure  5.5  Schematic of G r a n u l e 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 c o u l d o c c u r b) showing lumped  amplifications  194  Expressing q A  = A(Bp  and  in matrix  - Gq)  q  form  (5.27) matrices  elements are zero).  =  (l+AG)  _ 1  ensure  that  Cerebellar  Model  yields,  :  B are diagonal  diagonal  To  this  and  G  Solving  i s anti-diagonal for q  (ie.  i n terms of p  ABp q  the  yields,  (5.28)  i s an  orthonormal  discussed  in chapter  basis 2  set,  i s used.  the We  Pointwise thus  desire  that, q  = E 5  (5.29)  where, E  n  = D  -1/2 ,  /  (5.30)  z  n  n-1 D = l / n L p . p . /=0 ' '  (5.31)  T  n  Therefore  we  (I+A or  n  G  must )  n  _ 1  that,  B = E = D " n n n n  1  /  (5.32)  2  o  equivalently, B  ~ A 1  n  An  " (I+A 1  n  After B  while  the n n  t  are as  % i j a  G ) = D n n n  adjustment  requirements  and  A  ensure  '  U  .. -  h  ,  /  (5.33)  2  n  scheme using  adjustment,  has only  been locally  found  that  available  the elements of  meets  these  information.  the matrices  G ,  A  R  follows;  \l*ki*kj  (  5  '  3  4  )  0  (5.35)  0  (5.36)  mj b for  .  nij  i±j , g  .  =  and, =0  (5.37)  195  n  a  n/y  b for  -  }  (  -  5  1  3  8  )  (5.39)  Note  that  not learned.  thus  G  \l *ki*kjl~  w  0  i=j .  fixed, is  .  n /  [  =  we  a r e assuming  In matrix  that  notation  the  gains  the Golgi  ^  n  y  /  feedback  a  r  e  matrix  as f o l l o w s ,  n  l/n  =  X^*  T  "  A  n "  1  (  '  5  4  0  )  A:=0 It  c a n be c o n f i r m e d  desired First is  matrices we  equal B ~ n  Then  that  by s h o w i n g  c a n remove  A ~ n  this  1  ( I + A G ) n n  result  n "  1  (  I  +  A  n  6  n  )  =  A  n "  1  (  I  +  A  n  G  n  )  =  A  that  = A ~ n  n"'  of gains  equation  B  from  1 n  (5.33)  equation  results i s  in  the  satisfied.  (5.33)  as  i t  matrix.  c a n be  A  a choice  the matrix  to the identity 1  such  1  ( I + A G ) n n  (5.41)  simplified.  +  [  l  /  n  / A ^  T  " ^  ]  (  5  '  4  ^\ [h^  2  )  '  L  (5 43)  k=0 Substituting  A  n"  1  (  I  +  Let  us  they  appear  A  i n equation  n n> 6  assume  concerned  =  1/  that  n  (5.28)  ^ P  I  +  A  n  after  a  finite  equation  ( 5 . 4 4 ) a s n-°°  A  G  fixed  and  without  n  have  " '  }  A  A  values  introducing significant  r  number we  n  ]  and G  n  ( t o the degree  consider n  n  G  the matrices  as constants with)  (  yields,  T  converge  n  of precision of  c a n assume  error  into  5  that of  *  4  such that  adjustments.  f o r a l l terms  196  (  4  )  that we a r e If  we  the matrices  the  t h e sum. T h e  summation matrices  in  t h e summation  o . ~  A  1  (  l  +  c .  A  Rearrangement  A  The  " (l A 1  0 0  (I+A  G  )  o o  i s  l  /  n  ^ P * P *  T  a  - (I A G 1  c a  +  [  (  I  +  A  = O  G  C O > "  1  A  O O ]  a  o o  )]  (  T  =  T  5  -  4  5  >  (5.46)  I t i s not obvious  also  symmetrical.  L e t u s assume  Then  the transpose  i n the left  that  that  hand  the matrix  (I+A G  side  of  )  i s  (5.46)  yielding,  (I + A ^ G J ]  1  ]  be f a c t o r e d o u t .  yields,  G )[A  be removed  [A/  t  =  can thus  A^ i s s y m m e t r i c a l .  symmetrical. can  J  G  then  +  matrix  o f (5.44)  = }/n Lp p  2  n  k  (5.47)  T  k  k=0 Thus  i n the limit [A _  1  1  2  =  (5.48)  00 OO  ( I  + A G  OO  Recalling it  have,  " (I + A G ) ] OO  A  we  from  i s apparent We  OO  ) = D  /  = E  2  _  chapter  2 that  /  (5.49)  2  E ^ equals  the result  not  1  oo  CO  that  have  1  00 CO  actually  a scaler  i s as desired. been  able  to  prove  that  adjustment s c h e m e w i l l w o r k a s we h a d t o a s s u m e c o n v e r g e n c e t h e m a t r i c e s A n a n d G n , a n d we h a d t o a s s u m e t h a t (I+Aoo G o) o symmetrical. to  learn  I f these  assumptions  the  cerebellum  perform  the  adjustments  gains  through  (5.39).  are adjusted q.Ak) z j 3  are true,  then  this of was  i ti s possible  orthogonalization.  In  (5.34)  Q,  multiple,_of  =  i t does  not appear  of the gains A more  as  plausible  J  1  J  1  197  shown  possible in  hypothesis  by e x p o n e n t i a l a v e r a g i n g  ( 1 - e ) g . . (Jfc-1 ) + e p . ( J f c ) q . U )  t o be  to  equations  i s that the  where, (5.50)  for  i*j,  and,  a .U)  =  f  for  /=/,  is  a  and  (l-e)a the  small  .  /  ) +  other  adjustment  p o s i t i v e gain  continually  but  adaptation  with  i s slow, These  when  learning  is taking  for  fiber  of  chosen  not.  we  exponential z.  used  f  randomly.  whether  the  It  thus  signals,  quite  the  as  adjustments gain  occur  e,  that  factor,  be  to  triggered  Purkinje  cells  e  might  analogous  also  before,  as  a to  low occur  indicated  activity.  orthogonalization  variables  (5.51)  behavior  might  at  learned  input were  place  simulation  or  small  adjustments  Through  averaging  These  r e s u l t i n g in  filter.  Climbing  a  ( k)  r e l a t i o n s h i p s are  factor.  such  pass  by  ep. (it ) q  have  has  work  also  terms  may  a  prove from  will  this  the  been  in p  useful  method  either  true  values  polynomial  found  are  to  work  as  of  a  of  terms  strictly  method  its validity  proposed  with  Succesive  generate  polynomial  apart  that  averaging. to  It  found  the in  p  regardless  positive  or  orthogonalizing  model  of  part  of  the  cerebellum. Several cerebellar the  problems  model.  resulting  carried  on  the  remain  First,  vector, parallel that  they  negative  values  over  a  more  that  ensures  not  clear  defending  such  i s not  fibers  albeit  A  using q,  nature,  bias.  in  may  finite  of  learned  strictly  the  that  q  -  (1,...,1)  a l l Golgi  positive.  through  offset scheme  i s orthogonal.  cells  synapse  198  are  represent  plausible orthogonalization  that  scheme  with  as  a  orthogonalization,  cerebellum  effectively range  this  Signals  unipolar positive  by  a  and  positive  would  be  Secondly, mossy  in  fibers  one it is such  that  each  Golgi  adjustment  of  A more being are  Golgi  on  = 1/n  ..  TUJ  b .. p  /*j , g  n  are  5.3.3  where  the  gains  '  (5  52)  (5.54)  =  [1/n  .. =  (5.55)  [1/n  gains  I  k=0  _, q  ]  (5.56)  J  n-1 Z p.  k=0  . ]  (5.57)  K l  are  a s s u r e d l y a d j u s t e d on  information. precisely found  method  can  strictly  not be  because to  work  found  positive,  and  This  plausible  method  the  cannot  q^. a r e  not  variant model  of  q  basis be  for learning  locally to  work,  positive.  It  f o r the  same  reason.  ensure  that  the  to  these  of  proven  strictly  in simulations,  for offsetting a  the  methods  may  then  q^. be  a  orthogonalization  in  i n v e r s e dynamics  of  cerebellum. INPUT V A R I A B L E In  the  i s one  adjustments  0  .  functional the  signals  gain  -  0  however,  a  having  model.  (5.53)  available  If  in this  which  0  =  been  of  upon  and,  the  has  based  p^.  =  n i J  Now  signal  q 0  n i J  b  are  i n terms  accessible  n-1 a  gains  the  follows,  y  /  model  to  I? *;  g  for  access  feedback  locally  a d j u s t e d as  a  has  plausible  based  n/y  cell  order  two  portion  link of  the  to  SPLITTING  self-learn  the  Cartesian  manipulator,  we  had  manipulator's  reach.  199  to  restrict  T h i s was  ourselves  necessary  to  a  because  *  certain  functional  the  link  two  polynomials represents a  manipulator over  a  relationships i n the inverse could  t h e whole  limitation  n o t be  of the  imposed  kinematics  represented  as a  manipulator's  b y t h e u s e o f K-G  of  sum  reach.  of This  polynomials  as  basis set.  sum  As  mentioned  i n chapter  of  polynomials reach.  relationships  over  of more It  input  class  variable  that  variables  from  I t i s best  Consider  a  used  splitting, point  sum  of  piecewise  the  linear,  to the order  i n each  o f t h e sum  region.  Clearly  in  u s e a method  a  this  way.  we  call  that  t h e same e n d . we  mean  of view, of view  illustrated  of  the functional  c a n be m o d e l l e d  one c o u l d  the point  as  unique  portion  be m o d e l l i n g  according  to achieve  a control  use s e v e r a l  f o r each  space  of functions  variable  from  estimator.  thus  whole  etc.,  splitting  input  variable input  the  be p o s s i b l e  By  would  representations  general may  One  or q u a r t i c ,  polynomials  one c o u l d  representations  manipulator's  quadratic,  3,  by an  polynomials  that  a  single  i s encoded  o f t h e sum  on  of  input several  polynomials  example. representation  for  the  function, f over  = sin(27rz)  (5.58)  the range, -1  Clearly  < z < 1 a  (5.59)  fifth  order  sum  of  polynomials  representation  is  * required a  to achieve  a minimal  likeness  of f .  An  example  of  such  representation i s , sin(2;rz)  *  5.7z - 2 8 . 4 z  3  + 22.7z  200  5  (5.60)  which  was  needed  and  such  then a  Now  assume  4 z.  Note  saturated  as  Figure  5.7  each  sum  of  =* -  z  5.6  +  a  using  remains  to  when  variables  input such  functions have  polynomials.  reach, well  -  related  methods  It  are  It  ~  dynamics  of  and  likewise  z  represented  as  were  odd  1/4,  power  1/2,  variable  and z  not terms  1.  is  split  i s only  over  +  z  z. ,  z^. ,  be  this  based  the  to two  link  polynoials  over  of  that  the  better.  made  to  Efforts  the  learn  z.  Interference  be  on  ofsum  non-linear  s i t u a t i o n s where  201  two  manner.  overcome  i t possible  interest.  the  i s much  can  piecewise  of  represent  functions  whether  algorithms  whole  to  seen  splitting  estimates  of  as with  can  i t could  portions  (5.62)  determined  as  are  namely,  along  in  z^.  specific  2  It  split  variables  is possible  4  variable  other sums  It  sin(27rz)  make the  (5.61) the  2  3  using  in  the  z,  learning  warranted  over  to of  variables,  are  might  of  in  learning  of  of  terms  2i}}  varies.  2  2  equal  range  be  encountered  inverse  5  1.  input  or  we  +  the  plot  Minimization  of  z  representations.  representation  develop  z  shows  shows  at  polynomials 2  power  follows,  z.  1  even  coefficients  the  or  that  information  4z  of  that  polynomials  It  most -1  sin(27rz)  as  min{l,  either  a  Figure  the  variables  space  sin(27rz)  the  that  over at  noting  is achieved  = max{-1,  that  the  match  input  by  choosing  that  among  of  obtained  work to  estimates  limitations sums  the  of  K-G  Cartesian  manipulator's  functions the  are  not  region  of  Figure  5.6  Example of i n p u t v a r i a b l e  splitting  S  M . 0 0  Figure  5.7  -0.60  -0.20  R e p r e s e n t a t i o n s of  0.20  0.60  sin(2jrz)  202  J.00  With input more  regards  variables plausible Consider  point the by  of  muscles  they  to  assume  increase  over  on  taken  form  paired  the  some  are  fiber  as  resemble  on  the  sensors.  at  a  time  network  in  the  as  they  cerebellum,  functions  that  in  Consider  are  and  to  the  from are  Mossy  multiplied  result probably  assume  others  would maximum  that  some  high  levels  would  change  the  joint  angle,  If  these  variables  the  products  5.9.  Some a r e a  parameter  signals  203  to  figure  now  coming  in  minimum  together,  functions.  these  reasonable  others  where  spline  If  muscles  movement.  shown  signals  various  numerous i t is  at  encoded  and  variables.  a  say  the  saturated  while  example  which  A  control  angle,  in  are  reasonable  joint  a  effectively  its  those  in  joint  decrease  multiplied  wide.  a  be  l e v e l s to  fashion.  from  extension  from  input  and  scenario  five  basis  such  i s encoded  muscle  time  be  that  following.  opponents,  would  simple  described  there  movement  different a  or  into  of  i t is unlikely  cerebellum  can  Since  low  ranges  rather  physiological  or  at  the  joint  i t is  joint  shows a  functions products  of  large  several two  from  the  could  rotated  Also  change  5.8  these  angle  signals  joint  ranges  over  Figure  are  generally  would  similarly  encoded  joint.  some  to  in  tension  the  that  narrow  the  the  displacement.  signals  of  Model,  previously  variable  representing  as  the  variable  such  upon  in  is outlined  position  are  Cerebellar  split  input  One  acting  and  angular  an  The  signals  are  the  scenario  view.  elbow.  to  how  narrow  and  taken Fiber  realistic as  perhaps  a  joint  thousands  of  randomly,  four  Granule  Cell  -  together,  contains  Note  more such  is  many  the  set  of  spline-like  functions. As  This  pointed  desirable  more  such  general  spline  such  splines  spline  render  of  t h e model  Model,  into  more  would  are  well  generality.of learned. assumption  that  by  modified  Klett  sensory  physiological  a  required  free  the  being  K-G  Model  Klett  Model  that  evidence.  204  than  input  versus spline  functions gain can  the be  implausable space  variables.  be b a s e d  more  Albus  previous  that  i n i t s input  would  the  i t s  on t h e  by  advantage  Use o f  of functions  active  i s  the  i t would  i s based  would  proposed  for estimating  point  in contrast,  Model  variables  from  many  Incorporation  Model.  Model  i n terms  discernable  encoding  retain  Many  however,  model  continuous  i n the Albus  set of maximally Model,  would  of  effectively  Cerebellar  by p o l y n o m i a l s ;  Albus  to  coefficients  of  learning  polynomials.  i n order  Klett  have  estimates  i n the cerebellum.  applicable  Model  the each  data  t o form  t o be e s t i m a t e d ,  weighting  variables  the Albus  encoded  of  Klett  represented  Finally,  functions  s i m i l a r to the previous  i t s use  disadvantage^ of only  spline  minimizing  than  a modified  fewer  of  binary  functions  be  however.  c a n be u s e d  be g e n e r a t e d  many  because  essentially  that  may  tested,  of  of functions  The m o d i f i e d  requiring  terms  of the functions  could  t o be [26],  functions  class  functions  Albus[l].  Klett in  functions  t h e domain  remains  by  Spline  span  of  out  properties  interference. a  hypothesis  on a  consistent  is The  pattern with  F i g u r e 5.8  Encoding 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  Z (3) *Z  0.20  0.40  (4)  Z (5) *Z (6)  0.60  NORMALIZED JOINT ANGLE  Figure  5.9  Example  representing joint  of  products  angle  205  of  0.60  split  1.00  input  variables  BIBLIOGRAPHY  [I]  J.S. Albus, Theoretical C e r e b e l l a r Model, Ph.D. December 1972.  and Experimental Aspects of a Thesis, U n i v e r s i t y of Maryland,  [2]  J.S. Albus, Brains, Behavior & Robotics, 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.J Allum, "A L e a s t M e a n S q u a r e s L i n e D i f f e r e n t i a l of Sampled A n a l o g Comp., C - 2 4 , p . 5 8 5 , J u n e 1975.  [4]  R.B. Asher, D. A n d r i s a n i I I Adaptive Control Systems", p . 1226, A u g u s t 1976.  [5]  P.R. 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[58]  ACSL User Guide/Reference Manual, A s s o c . , I n c . , Concord, Mass.  210  Mitchell  and  Gauthier,  APPENDIX  A N A L Y T I C A L  K I N E M A T I C S  T H E  T H E  D I R E C T  From  Xj x  2  L I N K  TWO  L I N K  M A N I P U L A T O R  M A N I P U L A T O R  figure  the d i r e c t  position  kinematics  can  be  obtained,  =  l j S i n ( a j ) .+ l s i n ( a + o )  «=  -ljcostttj) -  2  Differentiation velocity  T H E  K I N E M A T I C S  the  immediately  TWO  O F  A  of  1  2  ljcosta^oj)  these  relationships  yields  kinematics, ljcosfaj)  • l^os^+aj)  , ljCOsCa^Oj)  ljSinUj)  + l sin(a +o )  ,  2  1  2  ljSinUj+Oj)  w h i c h c a n be r e a r r a n g e d a s , IJCOSUJ)  ,  l cos(a +a )  ljSinfaj)  ,  ^sinUj+aj)  2  t  211  2  a  1  + a  2  the  direct  and  represented as,  - -  —  "  M  1  _ 1 2_  _*2_  The the  —  d  +d  d i r e c t v e l o c i t y k i n e m a t i c s c a n be d i f f e r e n t i a t e d direct acceleration x  1  ,  l cos(a +a )  l sin(a )  ,  l sin(a +a )  -l sin(a )  , -ljSinCa^aj)  1  1  l  1  1  2  1  2  l cos(a ) l  yield  kinematics,  l cos(a ) 1  to  1  2  1  a  2  a  ljcosft^+a,,)  1  1 +  a  2  • 2 1  (  d  1  +  d  2  V  w h i c h c a n be r e p r e s e n t e d a s ,  - -  M  +  1  _ i a  -I  "1  M,  (d +d )2  2.  + a  1  2  THE INVERSE KINEMATICS Referring applied  again  (X -l 2  2  ( X - l 2  2  -l  2 l  2 2  c a n be  )/(-l l ) 1  2 1  - l  ) / l l  2 2  1  2  2  By t h e r u l e o f P y t h a g o r a s we have X  2Z  =  X  2 ^  +  X  that,  2 2  thus, cos(a ) = 2  or  rule  thus, cos(a ) =  and  the cosine  to yield,  cosU-a ) = and  t o the f i g u r e ,  (x  2 1  +x  2 2  -l  2 1  -l  2 2  )/l l 1  2  equivalently, a  2  = arccos((x  2 1  which i s the inverse The  inverse  2  2  2  +x -1 -1 J / ^ l ^ 2  1  2  p o s i t i o n kinematics function  position  kinematics function  212  for a . 2  for a  1  c a n be  obtained  by n o t i n g  that,  tanU) = ( l s i n ( a ) ) / ( l + l c o s ( a ) ) 2  2  1  2  2  and, tan(tf) =  x^/(-x ) 2  Since, o  1  = ^ - <t>  t h i s means t h a t , a  1  = arctan(x ,-x ) 1  Note t h a t  -  2  the arctan  arctan(l sin(a ),1 +l cos(a )) 2  2  1  2  2  f u n c t i o n u s e d h e r e h a s two a r g u m e n t s a n d i s  t h u s assumed t o y i e l d a r e s u l t i n t h e r a n g e -TT t o TT. An  alternative  kinematics  fora  representation  c a n be o b t a i n e d  1  of  the  by n o t i n g  c o s ( a ) = cos(\/>-0) = cos(\//)cos(0) 1  inverse  position  that,  + s i n (\p) s i n (4>)  By i n s p e c t i o n o f t h e f i g u r e we h a v e , cos(^) = -x /X 2  and, sin(^) = X j / X A p p l i c a t i o n of the cosine cosU) =  (X +f 2  2 1  -l  2 2  rule )/2l X 1  A p p l i c a t i o n of the sine r u l e sin(#)/l  yields,  yields,  = sin(ir-a )/X = sin(a )/X  2  2  2  and hence, sin(0) = l s i n ( a ) / X 2  Substituting costa^  2  these expressions  i n t o our previous  expression f o r  yields,  cosU,) = - x ( X + l 2  2  2 1  -l  2 2  )/2l X 1  2  + x l sin(a )/X 1  2  2  2  2 Subsitituting  i n the previous  expression  213  forX  and  combining  terms t h a t a r e equal t o c o s ( o ) y i e l d s , 2  cosfc^)  = [x l sin(a > 1  1  2  a n o t h e r form of t h e i n v e r s e  - x ( l + l c o s ( a ) ) ]/(x 2  1  2  2  kinematics function  2 1  +x  2 2  )  of a . 1  From t h e 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 h a v e , x  1  1  = M  X .  w h i c h c a n be r e a r r a n g e d t o y i e l d  the inverse  v e l o c i t y kinematics  as, = M -1  "1  d or  1 +  d  1  2  equivalently, d  a  l sin(a +a )  1  2  1  + a  1  2  , -^cos ( + a  2  l  2  1  1  k i n e m a t i c s we h a v e ,  x  a  1  which  can  d be  1 +  o  +  M,  )  1  From t h e d i r e c t a c c e l e r a t i o n = M  2  l cos(a )  -l sin(a )  l l sin(a )  2  1  2  1  (a,+a )2 2  2  rearranged  to yield  the  inverse  acceleration  kinematics as, • 2 ""1 d or  1 +  -1 = M d  M  1  1>4  2  (d +a )2 }  2  2  equivalently, l sin(a +a )  0,  2  '1 1  2  l l sin(a ) 1  2  2  1  2  l cos(a )•  -l sin(o ) 1  1  1  -l^jcosfaj) l l sin(a ) 1  2  , -ljcosft^+a.,)  ,  2  214  ,  -1^  ^ljCOSfdj)  1  . 2  L 2J  (d^dj)  X  APPENDIX B  ANALYTICAL DYNAMICS OF THE TWO LINK  MANIPULATOR  INVERSE DYNAMICS The L a g r a n g i a n [46,37] i s d e f i n e d as t h e 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 t h e p o t e n t i a l energy of a system, P, L = K - P The i n v e r s e dynamics e q u a t i o n s a r e o b t a i n e d a s , r. = 6_ [SL 1 - 3L 6t j a a j  The  da.  d e r i v a t i o n of t h e i n v e r s e dynamics e q u a t i o n s b e g i n s by  n o t i n g t h a t t h e k i n e t i c energy of mass m i s , 1  K  <= 1/2 m v  1  which  }  = 1/2 n ^ l ^ d ,  2  y  2  i s e q u a l t o 0 when n o t moving.  The p o t e n t i a l  energy  of  mass m i s , 2  P j » -n^gh « - m g l c o s ( a ) 1  1  1  which i s e q u a l t o 0 when x « 0 , 2  i e . a t the height of the o r i g i n .  The k i n e t i c energy of mass m i s , 2 K « 1/2 n» 2 2  v  2  The  2  square o f t h e v e l o c i t y o f mass m  direct  velocity  can be o b t a i n e d from  2  k i n e m a t i c s e q u a t i o n s g i v e n i n appendix  the A  by  noting that, v  2 2  -  i*  + x  2 2  and hence, v  2 2  « l, *, "* l 2  2  2 2  (d  2 1  +2d a +d 1  2  2 2  Thus t h e k i n e t i c energy o f mass m i s , 2  215  ) + 2l l o (a +d )cos(a ) l  2  1  1  2  2  K  = 1/2 n ^ l ^ d ,  2  + 1/2 m l  2  2  2  2  (d  +2d d +d  2  1  1  2  2  )  2  2  + n ^ ^ ^ c o s f a ^(d,  +a^a ) 2  The p o t e n t i a l energy of mass m i s , 2  P  = m gh = - m g l ^ o s t a ^ ) - m g l c o s ( a + a )  2  2  Combining  2  terms  2  yields  2  1  the Lagrangian  2  of  the  two  link  manipulator, L =  (m +m )l d  1/2  2  1  2  +  2  1  ]  m^ (a, +2d, o  1/2  2  2  2  +  d  2  2  )  2  + m l l cos(a )(d 2  1  +  Ty.  2  +a^a )  1  +  2  (m +m )gl cos(o ) 1  2  1  l  m gl cos(a^+a ) 2  Various and  2  2  2  d e r i v a t i v e s of t h e L a g r a n g i a n need t o be  obtained  combined t o y i e l d t h e 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 The d e r i v a t i v e s a r e , 3L/3d  «= ( m + m ) l  1  1  2  2  d  1  + m l  1  2  + 2m l l cos(a )d 2  6/6t(bL/dd )  1  2  2  2  + [m l 2  + 2 2 m  1  1  d  2  2  m^ljCOsta^dj  + m l  2  }  d  +  1  = [(m +m )l,  }  2 2  + 2m l l cos(o )3d,  2  2  2  2  1  + nijl^jcostaj) ]d  2  2  2  2  2 - 2m l l sin(a )d d 1  2  3L/3a  2  2  = -(m +m )gl sin(a )  1  2  l  - n» l l sin(o )d  2  2  - m 9 2 1  1  l  1  2  s i n  ^ i a  1  + a  2  2  2  2^  Combining t h e s e d e r i v a t i v e s y i e l d s t h e i n v e r s e dynamics e q u a t i o n for torque T T  1  =  d  1  ,  11 1 £  +  d  l2 2 a  +  d  +  d  111 1 A  1l2 1 2 d  A  2  +  +  d  d  l22 2 A  121 2 1 A  d  2  +  d  1  where, d  n  d  1 2  d  = (n^+m^lj = 2 2 m  1  +  m  l22 " ~ 2 1 2 m  l  l  2  + m l 2  2 i^2 1  s i n  C O S  2 2  + 2m l l cos(o ) 2  ^ 2^ a  ^ 2^ a  216  1  2  2  d d  1  1  =  2  121  d  -m l l sin(a ) 2  1  2  2  -«>2 l 2  =  1  1  s i n ( a  2  )  = (m +m )gl sin(a )  1  1  2  1  Similarly, various o b t a i n e d and c o m b i n e d  +  l  m gl sin(a,+a ) 2  2  derivatives  to yield  2  of t h e L a g r a n g i a n need  the inverse  t o r q u e T . The d e r i v a t i v e s a r e , 2 2 9 L / 9 d = n» 2 1 2 2 2 2 1 2  dynamics  t o be  function  for  2  1  2  d  + m  1  d  +  m  1  1  ^2 ^ 1  C 0 S  a  2  d  2  5/6t(3L/9d ) = [ m l 2  2  +  m  2  l 2  1 2  2 + m l 2 &2 ~ 2  9L/9a  2  = - m l l2Sin(a )d 2  Combining for  1  2  1  c o s  ^ 2^^1 a  2^i^2 ^ ^ 2^ 1 2  m  S  2  n  a  d  d  - n^l l sin(o )d d  1  1  2  2  1  2  m gl sin(a +a ) 2  2  1  2  these d e r i v a t i v e s  y i e l d s the inverse  dynamics e q u a t i o n  torque T , 2  r  2  = d  d  2 l  + d  1  +  o + d 2!2 1 2  2 2 d  2  d  d  d + d 221 2 1 2  2  n +  1  d  d  d  2 +  2  2 d  d 2  2 2  where, d  2 1  = m l  d  2 2  =  d  2l1  2  m l 2  2  222 *  0  d  2l2  0  d  221 2  * =  1  2  2  0  =  m gl sin(a +a ) 2  2  This  derivation  used  by  Paul  slightly  from  error  2 2  m l l sin(o )  =  d  d  ^ l ^ c o s U . , )  2 2  1  2  of the inverse  [46]. those  Note,  dynamics  however,  follows  that  the  g i v e n by P a u l a s t h e r e i s  in his derivation.  217  the  approach  results an  differ  arithmetic  APPENDIX  C L O S E D  U S I N G  T H E  L O O P  C O N T R O L  S E L F - L E A R N E D  A F T E R  2 0 0 ,  4 0 0 ,  C  O F  C A R T E S I A N  6 0 0 ,  T R A I N I N G  218  8 0 0  P A T H S  L I N E  1  I N V E R S E  A N D  1 0 0 0  D Y N A M I C S  0.00  S  0.60  4.00  1.60  T SEC  S  o  UJo COB ^d-  COB  CNJ  =>§  =>§  0.00  0.80  2.40  1.60  3.20  4.00  T SEC  in  8  o"  4.00  1.60  T SEC Closed  l o o p c o n t r o l of l i n e 1 u s i n g the s e l f - l e a r n e d  i n v e r s e dynamics a f t e r 200  t r a i n i n g paths  219  Cartesian  8 CM  8 CM  M—1  COo  r — i  CM  0.00  8  0.80  1.60  T SEC  2.40  3.20  4.00  8  4.00  Closed  l o o p c o n t r o l of l i n e 1 u s i n g the s e l f - l e a r n e d  i n v e r s e dynamics a f t e r 400  t r a i n i n g paths  220  Cartesian  s  s  CM • • CJ LU COS  CM CJ  0  COo  CM  6.00  8  0.80  J'SEC  0.00  0.80  1.60 T SEC  0.00  0.80  2 , 4 0  3.20  4.00  8  CJ  4.00  2.40  8  8  X oin 1.60 T  Closed  _  T.W  3.20  4.00  SEC  l o o p c o n t r o l of l i n e 1 u s i n g the s e l f - l e a r n e d  i n v e r s e dynamics a f t e r 600 t r a i n i n g p a t h s  221  Cartesian  S  8  OJ • •  CJ COo  2R  0.00  8  0.80  1.60  T SEC  2.40  3.20  4.00  8  CJ  3°  4.00 o m  8 o'"  Xo  OT  Closed  0.00  0.80  1.60  T SEC  2.40  3.20  4.00  l o o p c o n t r o l of l i n e 1 u s i n g the s e l f - l e a r n e d  i n v e r s e dynamics a f t e r 800  t r a i n i n g paths  222  Cartesian  8  8  CM  UJo  0.00  8  0.80  T SEC  3.20  4.00  8  CJ UJo CO 5  >8  4.00  in  8  Xo in  Closed  l o o p c o n t r o l of l i n e 1 u s i n g the s e l f - l e a r n e d  i n v e r s e dynamics a f t e r 1 0 0 0 t r a i n i n g p a t h s  223  Cartesian  APPENDIX  V I E W U S I N G  T H E  O F  S E L F - L E A R N E D  A F T E R  2 0 0 ,  2 2 0 0 ,  2 6 0 0 ,  4 0 0 ,  6 0 0 ,  3 0 0 0 ,  D  C I R C L E  D I R E C T 8 0 0 ,  3 4 0 0 ,  T R A I N I N G  224  1  ' P O S I T I O N 1 0 0 0 ,  3 8 0 0 , P A T H S  K I N E M A T I C S  1 4 0 0 ,  4 2 0 0 ,  1 8 0 0 ,  A N D  4 6 0 0  View  of  circle  1  k i n e m a t i c s a f t e r 200  using  the  self-learned  t r a i n i n g paths  225  direct  position  8  CM in  2Eo* • X UJCO  s  View  •0.16  -0.48  -0.80  of  circle  0.16  LEX CI)  1  using  M  the  self-learned  k i n e m a t i c s a f t e r 400 t r a i n i n g p a t h s  226  0.80  0.48  direct  position  View  of  circle  1  using  the  self-learned  k i n e m a t i c s a f t e r 600 t r a i n i n g p a t h s  227  direct  position  View  of  circle  1  using  the  self-learned  k i n e m a t i c s a f t e r 800 t r a i n i n g p a t h s  228  direct  pos  View  of  circle  kinematics a f t e r  1 1000  using  the  self-learned  t r a i n i n g paths  229  direct  position  o  (SI O" I  CO  O" I  (SIX • LU  CO t  -0.48  ).80  View  0.16  -0.16  0.48  0.80  LEX(l) M  of  circle  kinematics a f t e r  1 1400  using  the  self-learned  t r a i n i n g paths  230  direct  position  o  CM  O" I  CM tn  oo  CM—  9  S 1  .80  View  -0.48  of  circle  kinematics after  JTT  1  -0.16 LEX(I)  1 1800  using  0.16  0.48  0.80  M  the  self-learned  t r a i n i n g paths  231  direct  position  LEX(l) M  View  of  circle  1  using  the  self-learned  k i n e m a t i c s a f t e r 2200 t r a i n i n g p a t h s  232  direct  position  8  on  CM  in I  OD  o' I  CM— LU  K80  View  :  of  (M8  circle  ^OAB LEX(l)  1  using  OAS  0.80  (M8  M  the  self-learned  k i n e m a t i c s a f t e r 2600 t r a i n i n g p a t h s  233  direct  position  View  of  circle  1  using  the  self-learned  k i n e m a t i c s a f t e r 3000 t r a i n i n g p a t h s  234  direct  position  8 •  View  of  circle  1  using  the  self-learned  k i n e m a t i c s a f t e r 3400 t r a i n i n g p a t h s  235  direct  position  t  O"  o  xT LU  s  View  -0.16  -0.48  0.80  of  circle  ^  0.16  the  self-learned  LEX(1) M  1  using  0.48  k i n e m a t i c s a f t e r 3800 t r a i n i n g p a t h s  236  —i  0.80  direct  position  8  LEX(l)  View  of  circle  1  using  M  the  self-learned  k i n e m a t i c s a f t e r 4200 t r a i n i n g p a t h s  237  direct  position  View  of  circle  1  using  the  self-learned  direct  position  k i n e m a t i c s a f t e r 4600 t r a i n i n g p a t h s ( f i n a l 400 w i t h smoothing)  238  

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