BEHA VIOURAL ANALYSIS OF BIPEDAL ROBOT A dissertation submitted to the Department of Electrical Engineering, University of Moratuwa in partial fulfilment of the requirements for the Degree of Master of Science by SAMAN MANJULA WELIHINDA Supervised by: Prof. Lanka Udawatta Department of Electrical Engineering University of Moratuwa, Sri Lanka 2010 94542 Abstract In this thesis a 7 OoF bipedal robot has been simulated, and analyzed its behavior by varying torso angle to achieve dynamic stability while walking on sloping surfaces. The revolutionary dynamic stability criteria introduced by Prof. Miomir Vukobratovic in 1969 has been used throughout the thesis. The dynamically stable human walking on slopes are very complex and this thesis address this problem by starting from lower body and in the middle by adding a torso with the intention of gaining dynamic stability and finishes showing the effectiveness of the variation of torso angle. The ZMP formula presented in the paper of Prof. Miomir Vukobratovic is involved with huge computations and due to that most of the researches in this field have chosen alternatives such as GA and NN to find the ZMP. Although it is convenient to use GA and NN to avoid mathematical calculation the accuracy of resultant ZMP is questionable. The original ZMP equation has been used extensively in all cases to calculate ZMP in this thesis and rather verifies the accuracy of this concept. The simulations have been performed to verify the accuracy of the kinematics models that has been created before moving to the ZMP calculations. The effectiveness of the variation of torso angle on the dynamic stability of the bipedal robot has been analyzed by varying slope angle, step length and step time. It is indicated using real ZMP calculations the bipedal robot can maintain its dynamic stability while walking on slopes at any circumstances. D E C L A R A T I O N T h e w o r k s u b m i t t e d i n t h i s d i s s e r t a t i o n i s t h e r e s u l t o f m y o w n i n v e s t i g a t i o n , e x c e p t w h e r e o t h e r w i s e s t a t e d . I t h a s n o t a l r e a d y b e e n a c c e p t e d f o r a n y d e g r e e , a n d i s a l s o n o t b e i n g U ) J l l ' u r r e n t l ) s u b m i t t e d f o r a n y o t h e r d e g r e e . : ~­ ~·· S a m a n M a n j u l a W e l i h i n d a D a t e - 0 8 - 0 2 - 2 0 I 0 1 e n d o r s e t h e d e c l a r a t i o n b y t h e c a n d i d a t e . : j l . I , / ' t i ' P r o f L a n k a U d a w a t t a l ) c c l a r u t i o n \ b s t r a c t \ c k n m \ l e d g c m e n t I i s t o i ' • \ b b r e v i a t i o n s L i s t o l · I i g u r e s I J s l C l l . I ~1bles C h a p t e r s I . I n t r o d u c t i o n I . I B i p e d a l R o b o t C o n t e n t s I . 2 B a s i c s o f b i p e d a l r o b o t w a l k i n g 1 . 2 . 1 C a i t p h a s e s 1 . 2 . 2 S t a t i c a n d d y n a m i c w a l k i n g I . 2 . 3 Z e r o m o m e n t p o i n t I . 3 R e v i e w a n d r e s e a r c h I A M o t i v a t i o n 1 . 5 R e s e a r c h o b j e c t i v e I . 6 O v e l ' \ i e \ v 2 . B i p e d a l R o b o t D e s c r i p t i o n a n d T r a j e c t o r y P l a n n i n g 2 . 1 B i p e d a l R o b o t D e s c r i p t i o n 1 1 I r a j e c t o r y P l a n n i n g 2 . 2 . 1 T r a j e c t o r ) 2 . 2 . 1 . 1 J o i n t s p a c e t e c h n i q u e s 2 . 2 . 1 . 2 C a r t e s i a n s p a c e t e c h n i q u e s , . . ~ 2 . 2 . 2 S e l e c t i o n o f T r a j e c t o r i e s f o r t h e b i p e d a l r o b o t 2 . 2 . 2 . 1 T r a j e c t o r y f o r s w i n g l e g 2 . 2 . 2 . 2 T r a j e c t o r y f o r h i p 2 . 2 . 2 . 3 R e l a t i o n s h i p b e t w e e n s w i n g l e g m o v e m e n t a n d h i p m o v e m e n t 2 . 2 . 3 S e l e c t i o n o f j o i n t a n g l e t r a j e c t o r y f o r t h e b i p e d a l r o b o t J . 1 \ : . i n c m a t i c s o f t h e b i p e d a l r o b o t ) I K i n e m a t i c s o f t h e b i p e d a l r o b o t 3 . 1 . 1 K i n e m a t i c s o f t h e s w i n g l e g 3 . 1 . 2 C o o r d i n a t e c o n v e r s i o n 3 . 2 K i n e m a t i c s o f t h c s t a n c e l e g 3 . 2 . 1 J o i n t a n g l e s o f s t a n c e l e g 3 . 2 . 2 C o o r d i n a t e c o n v e r s i o n > . 3 C a l c u l a t i o n o f j o i n t a n g l e s 3 . 3 . 1 C a l c u l a t i o n o f j o i n t a n g l e s o f s w i n g l e g 3 . 3 . 2 C a l c u l a t i o n o f j o i n t a n g l e s o f s t a n c e l e g v V I V I I V I I I X 1 2 4 5 8 1 3 1 3 1 4 1 5 1 5 1 6 1 6 1 6 1 6 1 7 1 7 1 8 1 9 1 9 2 2 2 2 2 2 2 4 2 4 2 4 2 5 2 6 2 6 2 6 I I 3 . 3 . 3 M o d i f i c a t i o n o f j o i n t a n g l e s o f s w i n g l e g t o c o m p e n s a t e h i p 2 6 m o v e m e n t - L S i m u l a t i o n o f t h e b i p e d a l r o b o t 4 . 1 S i m u l a t i o n s o f t w a r e 4 . 2 S i m u l a t i o n o f s w i n g l e g 4 . 3 S i m u l a t i o n o f s t a n c e l e g 4 . 4 S i m u l a t i o n o f b o t h l e g s 4 . 5 S i m u l a t i o n o r l o w e r b o d y a n d t o r s o , • 2 8 2 8 2 8 3 2 " " _ ) - ' 3 6 5 . C a l c u l a t i o n o f Z M P a n d D y n a m i c s t a b i l i t y 3 9 5 . 1 G e n e r a l Z M P e q u a t i o n f o r t h e b i p e d a l r o b o t 3 9 5 . 2 D y n a m i c S t a b i l i t y M a r g i n ( D S M ) o f b i p e d a l r o b o t 3 9 5 . 3 A p p l i c a t i o n o f Z M P e q u a t i o n f o r t h e b i p e d a l r o b o t 4 0 5 . 3 . 1 M a s s c e n t r e c o o r d i n a t e s o f l i n k s 4 1 5 . 3 . 2 A n g u l a r a c c e l e r a t i o n o f l i n k s 4 1 5 . 3 . 3 A c c e l e r a t i o n o f l i n k s 4 2 5 . 3 . 4 I n e r t i a o f l i n k s 4 2 5 . 3 . 5 B u i l d i n g M A T L A B p r o g r a m t o c a l c u l a t e Z M P 4 3 5 . 4 C a l c u l a t i o n m e t h o d o f Z M P 4 3 5 . 5 / M P o f t h e l o w e r b o d y o r b i p e d a l r o b o t 4 4 5 . 6 Z M P v a r i a t i o n o f t h e l o w e r b o d y o f t h e b i p e d a l r o b o t 4 4 5 . 7 A d d i n g T o r s o 4 5 5 . 7 . 1 M a s s c e n t r e c o o r d i n a t e s o f t o r s o 4 5 5 . 7 . 2 A n g u l a r a c c e l e r a t i o n o f t o r s o 4 6 5 . 7 . 3 A c c e l e r a t i o n o f t o r s o 4 6 5 . 8 D e f i n i n g t o r s o a n g l e 4 6 ) C ) I M P o f t h e b i p e d a l r o b o t v v i t h t o r s o 4 7 " . 1 0 Z M P v a r i a t i o n o r t h e b i p e d a l r o b o t w i t h t o r s o 4 7 : ' . I I / M P o l ' t h c b i p e d a l r o b o t w i t h d i f f e r e n t t o r s o a n g l e s 4 8 5 . 1 2 Z M P v a r i a t i o n o f t h e b i p e d a l r o b o t w i t h d i f f e r e n t t o r s o a n g l e s 4 9 6 . B e h a v i o r a l a n a l y s i s o f b i p e d a l r o b o t 5 0 6 . 1 E f f e c t o f t o r s o a n g l e o n d y n a m i c s t a b i l i t y o n v a r y i n g s l o p e s 5 0 6 . 1 . 1 B i p e d a l r o b o t v . a l k i n g o n 5 d e g r e e s l o p e 5 1 6 . 1 . 2 B i p e d a l r o b o t w a l k i n g o n I 0 d e g r e e s l o p e 5 2 6 . 1 . 3 B i p e d a l r o b o t w a l k i n g o n 1 5 d e g r e e s l o p e 5 2 6 . 2 E f f ' c c t o f t o r s o a n g l e o n d y n a m i c s t a b i l i t y a t d i f f e r e n t s t e p s 5 3 6 . 2 . 1 B i p e d a l r o b o t v v a l k i n g o n f u l l s t e p 5 3 6 . 2 . 2 B i p e d a l r o b o t w a l k i n g o n h a l f s t e p 5 4 6 . 2 . 3 B i p e d a l r o b o t w a l k i n g o n q u a r t e r s t e p 5 4 ( J . 3 r ! T e e t o f t o r s o a n g l e o n d y n a m i c s t a b i l i t y a t d i f f e r e n t s t e p d u r a t i o n s 5 5 6 . 4 U l e c t o f t o r s o m a s s o n d y n a m i c s t a b i l i t y o f r o b o t 5 6 6 . 4 . 1 D y n a m i c s t a b i l i t y w i t h d i f f e r e n t m a s s e s o n t o r s o a t t o r s o a n g l e o f 5 6 ] ( ) ( ) I l l 6 . 4 . 2 D v n a m i c s t a b i l i t v w i t h d i f i C r c n t m a s s e s o n t o r s o a t t o r s o a m . ; l c o f 5 7 1 . : : ( ) . ~ ) 6 . 5 E f f e c t o n l e n g t h o f ' t o r s o o n d y n a m i c s t a b i l i t y o f r o b o t 5 7 6 . 6 E J T e c t o n w e i g h t s o f l o w e r b o d y o n d y n a m i c s t a b i l i t y o f r o b o t 5 8 6 . 7 B e h a v i o r a l a n a l y s i s o f j o i n t a n g l e s 6 0 7 . C o n c l u s i o n s a n d F u t u r e w o r k 6 2 7 . 1 C o n c l u s i o n s 6 2 7 . 2 F u t u r e W o r k 6 3 R e f e r e n c e s A p p e n d i c e s A p p e n d i x A M J \ T L A B p r o g r a m s A p p e n d i x B J o i n t a n g l e t r a j e c t o r i e s A p p e n d i x C : S i m u l a t i o n p r o g r a m s ~· 6 4 6 6 6 6 8 2 8 6 I V A c k n o w l e d g e m e n t III~lllb , t r c d u e p r i m a r i l ) t o 1 1 1 ) s u p e r v i s o r . P r o f e s s o r L a n k a U d m \ a t t a , f o r t h e c o n t r i b u t i o n ,1thi~ g t c < t t i i h i g h t s . p e r s p c c t i \ e s . c o r r e c t - g u i d a n c e a n d s e n s e o f h u m o r . I n a l l . I l l ) s i n c e r e t h a n k s g o t o t h e o f f i c e r s i n t h e P o s t G r a d u a t e O f f i c e . F a c u l t y o f l . n g i n e e r i n g . a n d t o t h e U n i v e r s i t y o f M o r a t u w a o f S r i L a n k a f o r t h e c o r p o r a t i o n e x t e n d e d t o m e i n n u m e r o u s \ \ a y s f o r t h e c l a r i f i c a t i o n o f m a t t e r s r e l a t e d t o m y a c a d e m i c s t u d i e s o n t i m e t i m H i g h o u t t h e c o u r s e . - ' .\l~o I l l ) s i n c e r e g r a t i t u d e e x t e m l e d t o t h e s t a f f w h o s e r v e s i n t h e D e p a r t m e n t o f E l e c t r i c a l E n g i n e e r i n g o f f i c e . I a l s o w i s h t o t h a n k m y f e l l o w e n g i n e e r s i n o f f i c e f o r t h e i r g e n e r o u s s u p p o r t g i v e n f r o m t h e b e g i n n i n g t o t h e e n d o f t h e c o u r s e . I a l s o c : - . . : t e n d m y s i n c e r e t h a n k s t o M r . A r u n a A b e y r a t n e f o r d e r i v i n g t h e e q u a t i o n s I l e e e s s < t i . \ f o r t h e s i m u l a t i o n s a n d c a l c u l a t i o n s . I s h o u l d t h a n k m a n : i n d i v i d u a l s . f r i e n d s a n d c o l l e a g u e s w h o h a v e n o t b e e n m e n t i o n e d h e r e p e r s o n a l ! ) i n m a k i n g t h i s e d u c a t i o n a l p r o c e s s a s u c c e s s . l~tst h u t n u t l e a s t I l l ) g r a t i t u d e g o e s t o m y w i f e f o r t h e l o v e , m o r a l s u p p o r t a n d u n d e r s t a n d i n g t i · o m s t a r t t o e n d o l ' t h i s c o u r s e . V I L i s t o f A b b r e v i a t i o n s a D o F g \ I T D - H r Z M P F R I F 7 r v i P I . I l l ( } C J . \ \ 1 \ C O M N P C M c , s , i f ( I ) S l o p e A n g l e D e g r e e s o f f r e e d o m A c c e l e r a t i o n d u e t o g r a v i t y S t e p l e n g t h S t e p T i m e D e n a v i t - H a r t e n b e r g n o t a t i o n L e n g t h o r l i n k i Z e r o m o m e n t p o i n t F o o t r o t a t i o n i n d e x F i c t i o n a l z e r o m o m e n t p o i n t L e n g t h o r T o r s o M a s s o i ' T o r s o T o r s o a n g l e C i e n e t i c a l g o r i t h m l \ e u r a l n e t \ \ o r k C e n t r e o r M a s s N o r m a l P r o j e c t i o n o f t h e C e n t r e o f M a s s C o s i n e o r j o i n t a n g l e i S i n e o f j o i n t a n g l e i J o i n t a n g l e a t t i m e t ~· V I I L i s t o f F i g u r e s F i g u r e 1 . 1 A n a t o m i c a l p l a n e s 1 . 2 L e g p o s i t i o n s d u r i n g o n e h a l f c y c l e 1 . 3 T h e c y c l e p h a s e r o t a t i o n o f b i p e d a l w a l k i n g 1 . 4 A b i p e d a l r o b o t s ' g a i t c y c l e 1 . 5 F o r c e s a n d t o r q u e a c t i n g o n a r o b o t f o o t 1 . 6 S i m p l i f i e d f o o t d y n a m i c s " ' 1 . 7 T h e b i p e d a l r o b o t s E O - E 3 i n t r o d u c e d b y H o n d a f r o m 1 9 8 6 - 1 9 9 1 1 . 8 i . 9 ! . I ( ) 1 . 1 1 1 . 1 2 2 . 1 I I I ~ - · · ' T h e b i p e d a l r o b o t s E 4 - E 6 i n t r o d u c e d b y H o n d a f r o m 1 9 9 1 - 1 9 9 3 I h e h u m a n o i d r o b o t s P I - P 3 p r e s e n t e d b y H o n d a f r o m 1 9 9 3 - 1 9 9 7 I h e h u m a n o i d r o b o t A S I M O l l u m a n o i d " ' Q R I O " i n t r o d u c e d b y S o n y c o m p a n y M I T h u m a n o i d r o b o t s : ( a ) S p r i n g F l a m i n g o . ( b ) S p r i n g T u r k e y . ( c ) ( l e e k . B o t 7 l i n k . 7 D o l - b i p e d a l r o b o t s , , i n g l e g a n d H i p t r a j e c t o r i e s A n g u l a r v e l o c i t y v a r i a t i o n o f t h e s w i n g l e g a n d s t a n c e l e g j o i n t a n g l e s f o r h a l f C \ C l e d u r a t i o n 3 . 1 S \ \ i n g l e g m o t i o n o f b i p e d a l r o b o t 3 . 2 S t a n c e l e g m o t i o n o f b i p e d a l r o b o t 3 . 3 T h e m o t i o n s o f s v · , i n g l e g a n d s t a n c e l e g - 1 . I F o o t t r a j e c t o r y a t 5 d e g r e e s l o p e - 1 . 2 I n i t i a l p o s e o f t h e S \ \ i n g l e g s i m u l a t i o n I ' s . I . I . 4'~1ll 1 j . - t . . ' . ' ' 1 1 1 g e g s 1 m u a t I o n a t · _ c a t a p o i n t - 1 - 1 s , , i n g l e g s i m u l a t i o n a t 1 6 3 ' d d a t a p o i n t - f . : ' I i n a ! p o s e o f t h e S \ \ i n g l e g s i m u l a t i o n - 1 . 6 I h e s w i n g l e g f o l l O \ v i n g p r e d e f i n e d t r a j e c t o r y - 1 . 7 T h e i n i t i a l a n d f i n a l p o s i t i o n s o f t h e s t a n c e l e g - 1 . 8 T h e s t a n c e l e g f o l l O \ v i n g t h e p r e d e f i n e d t r a j e c t o r y - f l ) I n i t i a l p o s i t i o n o f l o \ \ e r b o d y - I . I U L m \ e r b o d y s i m u l a t i o n a t 3 3 ' 0 d a t a p o i n t - 1 . 1 1 I 0 \ \ e r b o d y s i m u l a t i o n a t 1 5 4 1 1 1 d a t a p o i n t - 1 . 1 2 l i n a ! p o s i t i o n o f t h e ] ( m e r b o d y 4 . 1 3 L O \ \ C r b o d y f o l l o w i n g f o o t a n d h i p t r a j e c t o r y 4 . 1 4 I n i t i a l p o s i t i o n o f t h e b i p e d a l r o b o t 4 . 1 5 S i m u l a t i o n a t 4 i 1 1 d a t a p o i n t 4 . 1 6 S i m u l a t i o n a t 1 4 2 1 1 0 d a t a p o i n t 4 . 1 7 T h e f i n a l p o s i t i o n o f m o t i o n 4 . 1 8 T h e S \ v i n g l e g . h i p a n d t o r s o m o v e m e n t ' 1 . 1 D ) n a m i c s t a b i l i t y s a f e m a r g i n 5 . 2 L O \ w r b o d y o f b i p e d a l r o b o t 5 . 3 / M J > v a r i a t i o n o f L c m e r b o d y o f t h e b i p e d a l r o b o t P a g e 2 ' " > _ ) 3 4 5 6 8 8 9 1 0 I I 1 2 1 5 1 7 2 0 2 2 2 4 2 7 2 9 2 9 3 0 3 0 3 1 3 1 " ' I .)~ 3 2 ' " > ' " > . ) . ) ' " ' ' " ' . 1 . 1 3 4 3 4 3 5 3 6 3 6 3 7 3 7 3 8 4 0 4 0 4 4 V I I I ~-+ 1 3 i p c d a l r o b o t w i t h T o r s o 4 5 5 . 5 I l l u s t r a t i o n o f t o r s o a n g l e 4 6 5 . 6 / ' v 1 P o f b i p e d a l r o b o t w i t h T o r s o 4 8 5 . 7 I M P v a r i a t i o n o ! 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' l ' e r e n t t o r s o a n g l e s 4 9 6 . 1 I M P v a r i a t i o n w i t h t o r s o a n g l e a t 5 d e g . s l o p e w i t h f u l l s t e p l e n g t h 5 1 6 . 2 1 ! \ 1 P v a r i a t i o n \ \ i t h t o r s o a n g l e a t I 0 d e g . s l o p e w i t h f u l l s t e p l e n g t h 5 2 6 . 3 / M P v a r i a t i o n w i t h t o r s o a n g l e a t 1 5 d e g . s l o p e w i t h f u l l s t e p l e n g t h 5 2 6 . 4 Z M P v a r i a t i o n w i t h t o r s o a n g l e a t 5 d e g . s l o p e w i t h f u l l s t e p l e n g t h 5 3 6 . 5 Z M P v a r i a t i o n w i t h t o r s o a n g l e a t 5 d e g . s l o p e w i t h h a l f s t e p l e n g t h 5 4 6 . 6 Z M P v a r i a t i o n w i t h t o r s o a n g l e a t 5 d e g . s l o p e w i t h g u a r t e r s t e p l e n g t h 5 4 6 . 7 Z M P v a r i a t i o n w i t h t o r s o a n g l e a t 5 d e g . s l o p e w i t h f u l l s t e p l e n g t h v a r y i n g 5 5 s t e p t i m e 6 . 8 I M P v a r i a t i o n k e e p i n g t o r s o a n g l e a t I 0 d e g . a n d v a r y i n g t o r s o m a s s 5 6 6 . 9 Z M P v a r i a t i o n k e e p i n g t o r s o a n g l e a t 1 5 d e g . a n d v a r y i n g t o r s o m a s s 5 7 6 . 1 0 Z M P v a r i a t i o n k e e p i n g t o r s o a n g l e a t I 0 d e g . a n d v a r y i n g t o r s o l e n g t h 5 8 6 . 1 1 Z M P v a r i a t i o n k e e p i n g t o r s o a n g l e a t I 0 d e g . a n d v a r y i n g t h e w e i g h t s o f l o w e r 5 9 b o d y 6 . 1 2 A n g u l a r p o s i t i o n . A n g u l a r v e l o c i t y a n d A n g u l a r a c c e l e r a t i o n o f j o i n t a n g l e 8 1 6 0 6 . 1 3 A n g u l a r p o s i t i o n . A n g u l a r v e l o c i t y a n d A n g u l a r a c c e l e r a t i o n o f j o i n t a n g l e 0 2 6 0 6 . 1 4 A n g u l a r p o s i t i o n , A n g u l a r v e l o c i t y a n d A n g u l a r a c c e l e r a t i o n o f j o i n t a n g l e 0 , 6 1 6 . 1 5 A n g u l a r p o s i t i o n . A n g u l a r v e l o c i t y a n d A n g u l a r a c c e l e r a t i o n o ! " j o i n t a n g l e 8 ( , 6 1 I X L i s t o f T a b l e s I a I l k : ; i l l - 1 1 pdrcunetcr~ o f S \ \ i n g k g : 1 . 2 D - 1 1 p a r a m e t e r s o f s t a n c e l e g - 1 - . 1 P a r a m e t e r s f o r t h e s w i n g l e g s i m u l a t i o n 5 . 1 P a r a m e t e r s f ( ) r t h e c a l c u l a t i o n o f Z M P o f l m v e r b o d y 5 . 2 P a r a m e t e r s f o r t h e c a l c u l a t i o n o f Z V I P o f b i p e d a l r ( 1 b o t w i t h t o r s o \ 3 P a r a m e t e r s f o r t h e c a l c u l a t i o n o f Z M P o f b i p e d a l r o b o t w i t h d i f f e r e n t t o r s o a n g l e s 6 . 1 P a r a m e t e r s f o r t h e c a l c u l a t i o n o f Z M P o f b i p e d a l r o b o t a t d i f f e r e n t s l o p e s 6 . 2 P a r a m e t e r s f o r t h e c a l c u l a t i o n o f Z M P o f b i p e d a l r o b o t a t d i f f e r e n t s t e p s 6 . 3 P a r a m e t e r s f o r t h e c a l c u l a t i o n o f Z M P o f b i p e d a l r o b o t a t d i f f e r e n t s t e p t i m e s 6 . 4 P a r a m e t e r s f o r t h e c a l c u l a t i o n o f Z M P o f b i p e d a l r o b o t w i t h d i f f e r e n t m a s s e s o f l o r s o 6 . 5 P a r a m e t e r s f o r t h e c a l c u l a t i o n o f Z M P o f b i p e d a l r o b o t w i t h d i t T e r e n t l e n g t h s o f I o r s u P a g e I " . c _ _ ) 2 5 2 8 4 4 4 7 4 8 5 0 5 3 5 5 5 6 5 8 ( J . ( J l ) a r a m e t e r s f o r t h e c a l c u l a t i o n o f / M P o f b i p e d a l r o b o t w i t h d i f f e r e n t w e i g h t s o f 5 9 ] ( m e r b o d v X