A Modular Network for Legged Locomotion

7/30/2019 A Modular Network for Legged Locomotion

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ELSEVIER Physica D 115 (1998) 56-72PHYS I CA

A m o d u l a r n e t w o r k fo r l e g g e d l o c o m o t i o nM a r t i n G o l u b i t s k y a , ,, I a n S t e w a r t b , P i e t r o - L u c i a n o B u o n o a , j . j . C o l l in s c

a D e p a r t m e n t o f M a t h e m a t i c s , U n i v e r s it y o f H o u s t o n , H o u s t o n , T X 7 7 2 0 4 - 3 4 7 6 , U S Ab Ma them at ics Ins ti tu te , Un ivers i ty o f Warwick , C oven try CV4 7AL , U K

c D e p a r t m e n t o f B i o m e d i c a l E n g i n e e r in g , B o s t o n U n iv er si ty , 4 4 C u m m i n g t o n S t r ee t , B o s t o n M A 0 2 2 1 5 , U S AReceived 30 April 1997; accepted 26 August 1997Communicated by C.K.R.T. Jones

A b s t r a c t

In this paper we use symmetry methods to study networks of coupled cells, which are models for central pattern generators(CPGs). In these models the cells obey identical systems of differential equations and the network specifies how cells arecoupled. Previously, Collins and Stewart showed that the phase relations of many of the standard gaits of quadrupeds andhexapods can be obtained naturally via Hopf bifurcation in small networks. For example, the networks they used to studyquadrupeds all had four cells, with the understanding hat each cell determined the phase of the motion of one leg. However, intheir work it seemed necessary to employ several different four-oscillator networks to obta in all of the standard quadrupedalgaits.

We show that this difficulty with four-oscillator networks is unavoidable, but that the problems can be overcome by usinga larger network. Specifically, we show that the standard gaits of a quadruped, inc luding walk, trot and pace, cannot all berealized by a single four-cell network without introducing unwanted conjugacies between trot and pace - conjugacies thatimply a dynamic equivalence between these gaits that seems inconsis tent with observations. In this sense a s ingle networkwith four cells cannot model the CPG of a quadruped. We also introduce a single eight-cell network that can model all ofthe primary gaits o f quadrupeds without these unwant ed conjugacies. Moreover, this network is modula r in that it naturallygeneralizes to provide models of gaits in hexapods, centipedes, and millipedes. The analysis of models for many-leggedanimals shows that wave-like motions, similar to those obtained by Kopell and Ermentrout, can be expected. However, ournetwork leads to a prediction that the wavelength of the wave motion will divide twice the length of the animal. Indeed, wereproduce illustrations of wave-like motions in centipedes where the animal is approximately one-and-a-halfwavelength long- motions that are consistent with this prediction. We discuss the implica tions of these results for the development of modularcontrol networks for adaptive legged robots. Copyright 1998 Elsevier Science B.V.K e y w o r d s : Hopf bifurcation; Symmetry; Control pattern generator; Gaits

* Corresponding author.0167-2789/98/$19.00 Copyright 1998 Elsevier Science B.V. All rights reservedP H S0167-2789(97)00222-4


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M. Golubitsky et al . /Physica D 115 (1998) 56-72 571 . I n t r o d u c t i o n

T h e d y n a m i c s o f n e t w o r k s o f c o u p l e d i d e n t i c a l c e l ls h a s b e e n s t u d i e d i n c o n n e c t i o n w i t h t h e p h a se r e l a ti o n sh i p so b se r v e d i n t h e g a it s o f l e g g e d a n i m a l s [ 4 - 8 , 2 6 , 3 1 , 3 2 ] a n d c r e a t u r e s su c h a s th e l a m p r e y [ 2 , 2 1 - 2 3 , 2 5 ] w h o sel o c o m o t i o n i s e f f e c t e d b y t r a v e l i n g w a v e s o f m u sc l e c o n t r a c t io n s . H e r e w e u s e t h e t e r m ' c e l l ' t o i n d i c a te s e l f -c o n t a i n e d su b c i r c u i ts o f t h e n e t w o r k o r , m o r e m a t h e m a t i c a l l y , a sy s t e m o f o r d i n a r y d i f f e r e n t ia l e q u a t io n s . I t i s w i d e l yb e l i e v e d t h a t a n i m a l l o c o m o t i o n i s g e n e r a t e d a n d c o n t r o l le d , i n p r o- t, b y a c e n t r a l p a t t e r n g e n e r a t o r ( C P G ) , w h i c h i s an e t w o r k o f n e u r o n s i n t h e c e n t r a l n e rv o u s s y s t e m c a p a b l e o f p r o d u c i n g r h y t h m i c o u t p u t. C u r r e n t n e u r o p h y s i o l o g ic a lt e c h n i q u e s a r e u n a b l e t o i so l a t e su c h c i r c u i t s a m o n g t h e i n t r i c a t e n e u r a l c o n n e c t i o n s o f c o m p l e x a n i m a l s , b u t t h ei n d i r e c t e x p e r i m e n t a l e v i d e n c e f o r t h e i r e x i s te n c e i s s t r o n g ( f o r re v i e w s , s e e [ 1 7 - 1 9 , 2 4 , 2 7 , 2 9 ] ) . I r r e sp e c t i v e o fp h y s i o l o g i c a l c o n s i d e r a t i o n s , C P G - l i k e n e t w o r k s o f c o u p l e d c e l ls ( e . g ., e l e c t r o n i c c i rc u i ts ) a r e a n a t t r a c t i v e o p t i o nf o r t h e d e s i g n a n d c o n t r o l o f l e g g e d r o b o t s , b e c a u s e t h e y p r o d u c e a v a r i e t y o f p h a se r e l a t io n sh i p s i n a s t a b l e a n dn a t u r a l m a n n e r .

K o p e l l a n d E r m e n t r o u t [ 2 1 - 2 3 ] h a v e d e m o n s t r a t e d t h a t a li n e a r n e t w o r k c o n s i s t in g o f m a n y c o u p l e d i d e n ti c a lc e l ls c a n g e n e r a t e t r a v e l in g w a v e m o t i o n s s i m i l a r to t h o se o b se r v e d i n t h e l a m p r e y , a n d so m e b i o l o g i c a l p r e d ic t i o n sm a d e b y t h i s m o d e l h a v e b e e n c o n f i r m e d [ 3 0 ] . A l t h o u g h a l a m p r e y h a s n o le g s , t h e p h a se r e l a ti o n sh i p s f o u n d i nt h e se w a v e p a t te r n s a r e s i m i l a r t o t h o se t h a t o c c u r i n a n i m a l s su c h a s c e n t i p e d e s a n d m i l li p e d e s . A s r e g a r d s c r e a t u r e sw i t h a sm a l l n u m b e r o f l eg s , C o l l in s a n d S t e w a r t [ 5 - 8 ] h a v e sh o w n t h a t s im p l e sy m m e t r i c n e t w o r k s o f c o u p l e di d e n t i c a l c e ll s p o s se s s n a t u r a l p a t te r n s o f o sc i ll a t io n w i t h t h e s a m e p h a se r e l a t i o n sh i p s a s m a n y o b se r v e d g a i ts i nb i p e d s , q u a d r u p e d s , a n d h e x a p o d s . T h e se o sc i l l a ti o n p a t t e r n s a r e d e t e r m i n e d b y t h e p e r i o d i c s t a t e s t h a t c a n a r is e b yH o p f b i fu r c a t i o n f r o m t h e t ri v i a l g a i t ' s t a n d ' , a n d t h e i r e x i s te n c e a n d s t a b i li t y c a n b e a n a l y z e d u s i n g t h e t h e o r y o fs y m m e t r i c H o p f b i f u r c a ti o n o f G o h i b i t s k y a n d S t e w a r t [ 1 4 ,1 6 ]. S i n ce t h e a s s u m p t i o n o f s y m m e t r y i s b a s ic t o o u ra p p r o a c h , w e d i s c u s s t h e a p p r o p r i a t e n e s s o f sy m m e t r y in a n i m a l g a i t s i n so m e d e t a i l a t th e e n d o f th i s s e c t io n .

B e c a u s e o f th e s y m m e t r y a s s u m p t i o n , w e s h a ll t h ro u g h o u t u s e t h e t e r m ' n e t w o r k ' t o m e a n a f i x ed s y s t e m o fc o u p l e d i d e n t i c a l c e l l s ( r e p r e se n t e d s c h e m a t i c a l l y b y c i r c l e s ) w i t h a f i x e d s e t o f c o u p l i n g s ( r e p r e se n t e d s c h e m a t i c a l lyb y a r r o w s ) . T h e c o u p l i n g s t h e m se l v e s d i v i d e i n t o c l a s se s a n d a l l c o u p l i n g s i n a c l a s s a r e a s su m e d t o b e i d e n t ic a l .( D i f f e r e n t c la s se s o f c o u p l i n g s a r e r e p r e se n t e d b y d i f f e r e n t a r r o w s t y l e s .) F o r e x a m p l e , t h e C P G s o f b i la t e r a l lysy m m e t r i c a n i m a l s a r e u su a l l y m o d e l e d b y b i l a te r a l l y sy m m e t r i c n e t w o r k s w h e r e t h e c o u p l i n g s c o n n e c t i n g c e l l so n t h e t w o s i d e s o f t h e C P G ( c o n t r a l a te r a l c o u p l i n g ) a r e a s su m e d t o b e i d e n t ic a l a n d t h e c o u p l i n g s b e t w e e n n e a r e s tn e i g h b o r c e l l s o n t h e s a m e s i d e o f t h e C P G ( i p s il a t er a l c o u p l i n g ) a r e a s su m e d t o b e i d e n ti c a l, b u t t h e c o u p l i n g si n t h e se t w o c l a s se s a r e a s su m e d t o b e d i f f e r e n t . A l t h o u g h i t m i g h t b e p o s s i b l e t o a r r a n g e f o r t h e c o u p l i n g s i nd i f f e r e n t c l a s se s t o b e t h e s a m e , b y a r r a n g i n g f o r c e r t a i n c o u p l i n g s t r e n g t h s t o b e e q u a l , w e v i e w t h i s e q u a l i t ya s d i f fi c u lt t o c o n t r o l a n d m a i n t a i n , s i n c e t h e d i f f e re n t c l a s se s o f c o u p l i n g r e p r e se n t d i f f e r e n t t y p e s o f p h y s i c a lc o n n e c t i o n s .

A s a c o n se q u e n c e , c e r t a in p r o p e r t i e s o f t h e n e t w o r k s , su c h a s t h e sy m m e t r i e s o f t h e n e tw o r k , a r e f i x e d e v e nt h o u g h w e d o n o t sp e c i f y a t th i s l e v e l o f a b s t r a c t i o n t h e p a r ti c u l a r sy s t e m o f O D E s t h a t re p r e se n t a s i n g l e c e l l o r t h ee x a c t f o r m o f th e c o u p l i n g t e r m s b e t w e e n c e l l s. T h u s t h e sp e c i f i c a ti o n o f a p a r t i c u l a r n e t w o r k s c h e m e f o r a C P G i sa m o d e l i n g a s su m p t i o n t h a t f ix e s a sp e c t s o f t h e n e t w o r k su c h a s it s sy m m e t r i e s .

W i t h t h i s v i e w o f n e t w o r k s i n m i n d , C o l l i n s a n d S t e w a r t sh o w t h a t su i t a b l e sm a l l n e t w o r k s c a n g e n e r a t e m o s to f th e p h a se r e l a t i o n sh i p s f o u n d i n q u a d r u p e d a l a n d h e x a p o d a l g a i ts . H o w e v e r , th e y d o n o t d e sc r i b e a n y s i n g l en e t w o r k s t r u c t u r e t h a t is c a p a b l e o f g e n e r a t i n g a l l o b se r v e d p a t te r n s . M o r e o v e r , i n so m e o f th e i r n e t w o r k s , p a t t e rn sc o r r e sp o n d i n g t o d is t in c t g a i ts a r e ' c o n j u g a t e ' i n t h e m a t h e m a t i c a l m o d e l - e q u i v a l e n t u n d e r a n e t w o r k sy m m e t r y- a n d h e n c e m u s t b i f u r c a t e t o g e t h e r a n d p o s se s s t h e s a m e s t a b i l i t y c h a r a c t e r i s t i c s . I t i s d i f f i c u l t t o s e e w h y ac o r r e sp o n d i n g C P G w o u l d p r e f e r e n t i a l l y s e l e c t o n e o f th e se c o n j u g a t e p a t t e r n s a n d su p p r e s s t h e o t h e r. F o r t h e ser e a so n s , i t is n e c e s sa r y t o s e e k a m o r e s a t i s f a c to r y n e t w o r k .


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58 M . G o l u b i t s k y e t a l . / P h y s i c a D 1 1 5 ( 1 9 98 ) 5 6 - 7 2I n t h is p a p e r w e d e sc r i b e a s i m p l e n e t w o r k o f c o u p l e d i d e n t ic a l c e l ls w h i c h c a n g e n e r a t e t h e f u l l r a n g e o f p h a se

r e l a t io n sh i p s o b se r v e d i n t h e g a it s o f 2 n - l e g g e d a n i m a l s , f o r a ll v a l u e s o f n . T h e s a m e sc h e m a t i c s t r u c tu r e a l sog e n e r a t e s t r a v e l i n g w a v e s o f t h e k i n d s o b se r v e d i n l e g l e s s a n i m a l s su c h a s e e l s , w o r m s , sn a k e s , a n d l a m p r e y .M o r e o v e r , i n t h is c i r c u i t th e r e a r e n o u n w a n t e d c o n j u g a c i e s , so p a t t e r n s o f o sc i l la t i o n t h a t sh o u l d b e c o n s i d e r e dd i s t in c t i n t h e a n i m a l a r e a l so d i s t in c t m a t h e m a t i c a l ly , a n d i n p a r t i c u l a r c a n b i f u r c a t e a t d i f f er e n t p a r a m e t e r v a l u e sf r o m e a c h o t h e r a n d c a n p o s se s s d i f f e r en t s t a b i li t y c h a r a ct e r is t i c s. I n a d d i ti o n , t h e s t r u c t u r e o f t h e n e t w o r k i sm o d u l a r : in a 2 n - l e g g e d a n i m a l t h e n e t w o r k h a s o n e ' h e a d ' m o d u l e , o n e ' t a i l ' m o d u l e , a n d n - 2 i d e n t i c a l ' b o d y 'm o d u l e s . S u c h a s t r u c tu r e h a s o b v i o u s u t i li t y i n ro b o t i c s . I m p o r t a n t l y , o u r n e t w o r k s c h e m a t i c s c a n b e r e a l i z e d i nm a n y d i f f e r e n t w a y s , a n d a s l o n g a s t h e sy m m e t r y is p r e se r v e d , t h e s a m e b a s i c r a n g e o f p h a se r e l a t i o n sh i p s w i l l inp r i n c i p l e b e g e n e r a t e d .

T h e c r u c i a l o b se r v a t i o n i s t h a t in o r d e r f o r a sy m m e t r i c n e t w o r k o f i d e n t i c a l c e l ls t o r e p r o d u c e t h e p h a se r e l a -t i o n sh i p s f o u n d i n g a i t s o f a 2 n - l e g g e d a n i m a l , t h e n u m b e r o f c e l l s sh o u l d b e ( a t l e a s t ) 4 n , n o t 2 n a s p r o p o se db y C o l l i n s a n d S t e w a r t . F o r i n s t a n c e , 8 c e l l s a r e n e e d e d f o r a q u a d r u p e d a n d 1 2 f o r a h e x a p o d . T h i s o b se r -v a t i o n l e a d s t o p r e d i c t i o n s a b o u t w a v e n u m b e r s t h a t p o t e n t i a l l y m i g h t d i s t i n g u i sh su c h a n e t w o r k f r o m l i n e a rc h a i n s , a n d t h e r e i s e v i d e n c e f r o m c e n t i p e d e g a i t s t o su p p o r t t h o se p r e d i c t i o n s , a p o i n t t h a t w i l l b e d i s c u s se db e l o w . W e n o t e t h a t E r m e n t r o u t a n d K o p e l l [ 1 2 ] c o n s i d e r c e l l n e t w o r k s w i t h 8 a n d 1 2 c e l l s t h a t a r e s i m i l a rt o o u r s . H o w e v e r , t h e f o r m o f th e c o u p l i n g i n t h e ir n e t w o r k s ( a n d t h e r e su lt i n g sy m m e t r i e s ) a r e d i f fe r e n t f r o mours .

1.1. Rat ionale fo r symm etryT h e a s su m p t i o n t h a t a C P G h a s so m e d e g r e e o f sy m m e t r y m a y s e e m r a t h e r a r b i tr a r y , b u t s e v e r a l d i f f e r en t p o i n ts o f

v i e w su g g e s t th a t i t i s b o t h n a t u r a l a n d r e a so n a b l e . W e p r o v i d e a f e w c o m m e n t s t o m o t i v a t e t h e sy m m e t r y a s su m p t i o n .B i l a t e r a l sy m m e t r y is p r e v a l e n t t h r o u g h o u t t h e a n i m a l k i n g d o m , so a b i l a t e r a ll y sy m m e t r i c C P G s e e m s v i r t u a l lym a n d a t o r y . A l t h o u g h t h e p h y s i o l o g y o f a n i m a l l i m b s d i f f e r s a c c o r d i n g t o th e p o s i t i o n o f th e l i m b - e sp e c i a l l y inq u a d r u p e d s , w h e r e f r o n t l i m b s t y p i c a l l y d i f f e r i n d e t a il f r o m b a c k l i m b s - t h e se d i f f e r e n c e s n e e d n o t e x t e n d t o t h eu n d e r l y i n g C P G . T h e r o l e o f t h e C P G i s to p r o v i d e t i m i n g s i g n a l s to t h e l i m b s , w i t h su i t a b l e p h a se sh i ft s : th e r e i sn o t h i n g i n t h e p a t t e r n o f t h o se p h a se sh i f t s t o i n d ic a t e sp e c i a l t i m i n g d i f f e r e n c e s b e t w e e n f r o n t a n d b a c k . I n d e e d t h eo b se r v e d p h a se sh i f ts i n m a n y g a i t s a r e s i m p l e f r a c ti o n s o f t h e g a i t p e r io d , f o r e x a m p l e 0 , , 1 , , i n t h e q u a d r u p e dw a l k a n d 0 , , ~ in t h e m e t a c h r o n a l g a i t o f a h e x a p o d . S u c h p h a se r e l a t io n sh i p s a r e c o m m o n i n sy m m e t r i c n e t w o r k sb u t u n e x p e c t e d i n a s y m m e t r i c o n e s .

I n d e e d m o s t o sc i l la t o r n e t w o r k m o d e l s o f C P G s a s su m e t h a t th e o sc i l la t o r s a r e i d e n ti c a l, w h i c h i s a t a c i t sy m m e t r ya s su m p t i o n . M o r e c r u c i a l t o o u r a p p r o a c h i s t h e id e a t h a t th e c o u p l i n g p a t t e r n s i n th e n e t w o r k a l so h a v e so m e d e g r e eo f sy m m e t r y . I t is p o s s i b l e , f o r e x a m p l e , t o u se v a r i a b l e c o u p l i n g s d e s i g n e d t o e n f o r c e p a r t i c u l a r p h a se r e l a t io n sh i p s .W e b e l ie v e , h o w e v e r , t h a t sy m m e t r y o f th e e n t i r e n e t w o r k i s a m o r e p l a u s i b l e m e c h a n i sm , f o r s e v e r a l r e a so n s . T h ef i rs t i s t h a t w h i l e i t i s o f c o u r se p o s s i b l e t o d e s i g n c o u p l i n g s t r e n g t h s t h a t e n f o r c e a n y r e a so n a b l e s e t o f p h a ser e l a t io n sh i p s , su c h m o d e l s d o n o t s e l e c t p a r ti c u l a r p h a se s a s b e i n g i n a n y w a y sp e c i a l . T h e s a m e d e s i g n p r i n c i p l ec o u l d e q u a l l y r e a d i l y p r o d u c e a w a l k w i t h p h a se o f , s a y , 0 , 0 .1 7 3 , 0 . 6 2 8 , 0 . 8 4 4 . Y e t t h e p h a se s a c t u a l l y o b se r v e da r e s i m p l e f ra c t i o n s o f t h e p e r i o d . I t is h a r d t o s e e w h y a d e s i g n p r i n c i p l e th a t c a n p r o d u c e any p h a se r e l a ti o n sh i p sw h a t so e v e r sh o u l d l e a d t o th e e v o l u t i o n o f a C P G w i t h s i m p l e f r a c ti o n a l p h a se s . I n sh o r t , w e w i sh t o explain th eo b s e r v e d p h a s e s , w i t h i n s o m e h y p o t h e t i c a l f ra m e w o r k , a n d n o t j u s t reproduce t h e m i n a C P G c i r c u i t . A n o t h e ra r g u m e n t i s p h y s i o l o g i c a l . F o r c r e a t u r e s w i t h s i x o r m o r e l e g s , t h e r e i s o f t e n l i t t l e o v e r t d i f f e r e n c e b e t w e e n o n ep a i r o f l i m b s a n d a n o t h e r p a i r . I n m o s t b e e t l e s , a l l s i x l e g s l o o k f a i r l y s i m i l a r . T h i s o b se r v a t i o n i s e v e n s t r o n g e rf o r a r t h r o p o d s , w h o se e v o l u t i o n a r y h i s t o r y r e s t s o n th e t r i c k o f d e v e l o p i n g l a r g e n u m b e r s o f m o r e o r l e s s id e n t i c a ls e g m e n t s ( w h i c h m a y l a t e r b e c o m e sp e c i a l i z e d i n so m e r e sp e c t s - e sp e c i a l l y a t t h e h e a d a n d t a il ) . ' I d e n t i c a l


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M . G o l u b i t sk y e t a l . / P h y s i c a D 1 1 5 ( 1 9 9 8 ) 5 6 - 7 2 59s e g m e n t s ' i s a s y m m e t r y s t a t em e n t : i t i m p l ie s t h a t ( m o d u l o t h e e n d s ) t h e a n i m a l h a s a t ra n s l at i o n a l s y m m e t r y . Itw o u l d t h e r e f o r e n o t b e s u r p r i s i n g t o f i nd t h a t i ts C P G h a s s i m i l a r ( p e r h a p s i m p e r f e c t ) t r a n s la t i o n al s y m m e t r y .

W e e m p h a s i z e t h a t t h e a b o v e c o n s i d e r a t i o n s a r e i n t r o d u c e d i n o r d e r t o m o t i v a t e o u r s y m m e t r y a s su m p t i o ns .O u r n e t w o r k c o u l d h a v e u s e f u l p o t e n t i a l i n r o b o t i c s e v e n i f i t d o e s n o t t u r n o u t t o m o d e l a n i m a l C P G s . H o w e v e r ,t h e c a s e f o r s y m m e t r y - o r a p p r o x i m a t e s y m m e t r y - i n r e a l C P G s s e e m s t o u s to b e p la u s ib l e . I t i s i m p o r t a n t t or e c o g n i z e , h o w e v e r , t h a t r e al s y s t e m s , e s p e c i a l l y b i o l o g i c a l o n e s , a r e n e v e r p e r f e c t l y s y m m e t r i c. T h i s d o e s n o t h a v ea n y s e r i o u s e f f e c t o n o u r a n a l y si s , f o r th e f o l l o w i n g g e n e r a l r e a s o n . T h e d y n a m i c s o f a s y s t e m t h a t i s ' c l o s e t o 's y m m e t r i c - p e r t u r b e d a w a y f r o m a n i d e a l s y m m e t r i c s y s t e m - i s i n m a n y r e s p e c t s v e r y s im i l a r to t h a t o f th e i d e als y s t e m . T h e l o s s o f p e r f e c t s y m m e t r y r e s u lt i n g f r o m a p e r t u r b a t i o n m a y c h a n g e f i n e d e t a il s o f th e d y n a m i c s ( f o re x a m p l e s t a t es t h a t w e r e p r e v i o u s l y r e l a te d b y a s y m m e t r y m a y b e c o m e d i st i n ct ) b u t t h e b r o a d p a t t e r n r e m a i n sm u c h t h e s a m e ( t h e d i f f e r en c e s b e t w e e n t h o s e s ta t es w i l l u s u a l ly b e s m a l l) . T h e d y n a m i c s o f a s y s t e m t h a t is c l o s et o s y m m e t r y r e s e m b l e s t h a t o f th e i d e a l s y m m e t r i c s y s t e m f a r m o r e c l o s e l y t h a n it r e s e m b l e s t h e d y n a m i c s o f a' g e n e r i c ' a s y m m e t r i c s y s t e m .

F o r t h e s e r e a so n s , w e b e l i e v e th a t s y m m e t r y i s a r e a s o n a b l e m o d e l i n g a s s u m p t i o n , a n d o u r i n te n t i o n w i l l b e t op u r s u e t h e c o n s e q u e n c e s o f m a k i n g s u c h a n a ss u m p t i o n . A s w i l l b e c o m e a p p a r e n t , t h e d y n a m i c s o f a s y m m e t r i cC P G i s r e m a r k a b l y c l o s e t o t h a t o b s e r v e d i n a c tu a l g a i t s - e v e n f o r ' e x o t i c ' g a i t s s u c h a s g a ll o p s a n d t h e c a n te r ,w h e r e a t fi r st s i g h t t h e s y m m e t r y p r o p e r t i e s o f p h a s e r e l a t i o n s h i p s a r e n o t e s p e c i a l l y a p p a r e n t .

2 . T h e o b s t r u c t i o n o f c o n j u g a t e s o l u t i o n sL e t C 1 , C 2 . . . . , C N b e n o m i n a l l y i d e n t i c a l c e l l s , a n d l e t t h e s t a t e v a r i a b l e s o f C i a t t i m e t b e r e p r e s e n t e d b y a

v e c t o r x i = ( x ~ . . . . . x i ) ~ N s , w h e r e N s is a n s - d i m e n s i o n a l E u c l i d e a n s p a c e . I n t h e li t e r a tu r e i t is u s u a l t o r e f e rt o s u c h a c e l l a s a n ' o s c i l la t o r ' , b u t w e p r e f e r t o a v o i d t h is t e r m i n o l o g y s i n c e s u c h c e l l s n e e d n o t b e c a p a b l e o fo s c i l l a t i n g a s i n d i v i d u a l s . T h e y m a y n e v e r t h e l e s s a c q u i r e t h e a b i l i t y t o o s c il l a t e w h e n t h e y a r e c o u p l e d i n n e t w o r k s .A n e a r l y e x a m p l e o f t h is p h e n o m e n o n i s d u e t o S m a l e [ 2 8 ] ; o t h e r e x a m p l e s c a n b e c o n s t r u c t e d u s i n g c e ll s y s t e m swi th s = 1 .

W e a s s u m e t h a t e a c h c e ll c a n b e m o d e l e d b y a n o r d i n a r y d i f f er e n ti a l e q u a t i o n ( O D E ) o f t h e f o r md x i - - f ( x i , ) ~) , (1 )d t

w he r e )~ = ( )~1 . . . . )~p) i s a vec to r o f p p a ram e te r s and f : Ns x NP - -+ Ns i s a s m oo th vec to r f i e ld . S in ce f i si n d e p e n d e n t o f i , e a c h c e l l h a s t h e s a m e d y n a m i c s a s m ay o t h e r a t g i v e n p a r a m e t e r v a l u e s , a n d th i s is t h e s e n s e inw h i c h t h e y a r e ' i d e n t i c a l '.

W e f u r t h e r a s s u m e t h a t t h e s e c e l l s a r e c o u p l e d i n a n e t w o r k , i m p l y i n g t h a t t h e s ta t e o f a g i v e n c e l l a f f e c ts t h es t at e s o f a ll t h o s e t o w h i c h i t is c o u p l e d . F o r s i m p l i c i ty o f d e s c ri p t i o n w e m o d e l u n i d i r e c ti o n a l c o u p l i n g o f C j t oC i b y a te r m o ~ i jh ( x J , x i ) , w h e r e o l i j i s th e s t r e n g t h o f t h e c o u p l i n g f r o m C j t o C i . F o r e x a m p l e , d i f f u si v e c o u p l i n gh a s t h e f o r m h ( x j , x i ) = x j - - x i . N o t i c e t h a t i f o t i j = O , t h e n c e l l C j h a s n o d i r e c t e f f e c t o n c e l l C i ; t h e y a r eu n c o u p l e d . B i d i r e c t io n a l c o u p l i n g w i l l b e r e p r e s e n t e d a s a c o m b i n a t i o n o f tw o e q u a l a n d o p p o s i t e u n i d i r e ct i o n a lcouplings o l i j = o l j i ) . A n a l t e r n a t i v e t o d i f f u s i v e c o u p l i n g i s s y n a p t i c ( p u l s e ) c o u p l i n g . W e s t r e s s t h a t o u r r e su l t sd o n o t d e p e n d o n t h e f o r m o f t h is c o u p l i n g , b u t o n l y u p o n t h e s y m m e t r i e s o f t h e re s u l t in g s y s t e m , s o m a n y o t h e re x p r e s s i o n s f o r t h e c o u p l i n g t e r m s c o u l d b e e m p l o y e d i n st e a d. I n p a r t i c u l a r w e c o u l d r e s tr i c t th e e f f e c t o f t h ec o u p l i n g to o n e o r m o r e c o m p o n e n t s o f x i a n d x j , f o r e x a m p l e t o m o d e l v o l t a g e - c o u p l e d n e t w o r k s .

G r a p h i c a l l y w e r e p r e s e n t s u c h a n e t w o r k a s a s et o f n o d e s j o i n e d b y l i n e s m a r k e d w i t h a r r o w s . C o u p l i n gf r o m C j t o C i i s r e p r e s e n t e d b y a l i n e o f s t r e n g t h o l i j w h o s e a r r o w p o i n t s to w a r d s C i , a n d c o u p l i n g s o f s t r e n g th


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60 M . G o l ub i ts lc y e t a l . / P h y s i c a D 1 1 5 ( 1 9 9 8 ) 5 6 - 7 2z e r o a r e s u p p r e s s e d . I n p r a c ti c e , c o n n e c t i o n s o f d i f f e ri n g s t r e n g t h a r e r e p r es e n t e d b y a r r o w s w i t h d i f f e re n tm a r k i n g s .

T h e n e t w o r k c a n t h u s b e r e p r e s e n t e d b y a c o u p l i n g m a t r i x A - = ( o l i j ) , w h e re 1 _< i , j _< N . T h e e q u a t io n s fo rt h e n e t w o r k b e c o m e

d x i--- f (x i , )~) - I- ~ - ~ o t i j h ( x j , x i ) . (2 )d t j # i

N o w w e a r e in a p o s i t i o n t o ad d r e s s t h e is s u e o f s p a t i o - te m p o r a l s y m m e t r i e s o f s u c h s y s t e m s o f O D E s . B y a s p a t i a ls y m m e t r y w e h e r e m e a n a p e r m u t a t i o n o f t h e c el ls , th a t i s, a b i j e c ti v e m a p p i n g o- : {1 . . . . . N } ~ {1 . . . . . N } ,w h i c h p r e s e rv e s t h e c o u p f i n g m a t ri x . T h e p e r m u t a t i o n ~r a c ts o n x = ( x l , . : . , x s ) b y

c r X = ( X c r - l ( 1 ) . . . . . X o . - l ( s ) ) .S p e c i f i c a l l y , w e l e t F b e t h e g r o u p o f p e r m u t a t i o n s c r s u c h t h a t

otcr(i),~r(j ) = lY i j f o r a l l i , j 6 {1 . . . . . N } . ( 3 )

I f w e n o w w r i t e E q s . ( 2 ) i n t h e a b b r e v i a t e d f o r md x - F ( x , ) 0 + H ( x ) - - g ( x , 9 0 ,d tt h e n t h e f u n c t i o n g w i l l b e F - e q u i v a r i a n t , t h a t i s , i t w i l l s a t i s f y t h e c o n d i t i o n

g ( c r x , ) ~ ) = c r g ( x , ) ~)f o r a ll o - 6 F . T h i s i m p l i e s t h a t w h e n e v e r x ( t ) i s a so lu t i o n , so i s c r x ( t ) . W e c a l l t h e s e c o n j u g a t e s o l u t i o n s .

A t e m p o r a l s y m m e t r y i s a p h a s e s h i f t o n a p e r i o d i c s o l u t i o n o f E q s . ( 2 ) . S p e c i f ic a l l y , i f x ( t ) i s a p e r i o d i c s o l u t i o nw i th p e r io d T , a n d q) c 5 1 - - R /7 7 --= [0 , T ) w i th 0 a n d T id e n t i f i e d , t h e n x ( t ) h a s s p a t i o - t e m p o r a l s y m m e t r y(a ,~0 ) E F x ~1 i f

a x ( t + ~ ) = x ( t ) (4 )f o r a l l t , o r e q u i v a l e n t l y

c ~ x ( t ) = x ( t - ~ o) . (5 )I n o t h e r w o r d s , a p p l y i n g t h e s p a t i a l s y m m e t r y o - h a s t h e s a m e e f f e c t a s s h i f t i n g p h a s e b y a f r a c t i o n ~0 o f a p e r io d .

T h e s y m m e t r y g r o u p o r i s o t ro p y g r o u p o f a p e r i o d i c s o l u t i o n x ( t ) i s th e g r o u p o f a ll s p a t i o - t e m p o r a l s y m m e t r i e so f x ( t ), c o n s i d e r e d a s a s u b g r o u p o f F x S 1 . C o n j u g a t e s o l u t i o n s h a v e c o n j u g a t e s y m m e t r y g r o u p s .

T h e t h e o r y o f s y m m e t r i c H o p f b if u r c a t i o n p r o v i d e s g e n e r a l c o n d i t i o n s t h a t g u a r a n t e e t h e e x i st e n c e o f p r i m a r yb i f u r c a t i o n s , f r o m t h e t ri v i a l s o l u t i o n x i = 0 f o r a l l i , t o p e r i o d i c o s c i l l a t i o n s w i t h p r e s c r i b e d s p a t i o - t e m p o r a ls y m m e t r i e s . T h e s y m m e t r y g r o u p s o f t h e r e s u l ti n g p a t te r n s a r e t h e s u b g r o u p s o f F x ~ 1 t h a t h a v e tw o - d i m e n s i o n a lf i x e d p o i n t s p a c e s , s e e [ 1 4 , 1 6 ]. M o r e r e c e n t l y s u c h s u b g r o u p s h a v e b e e n c a l l e d C - a x i a l s u b g r o u p s , s e e [ 9 , 1 0 ] .W e a r e n o w i n a p o s i ti o n t o s k e t c h a g r o u p - t h e o r e t ic a r g u m e n t s h o w i n g t h a t t h e r e is n o s i n g l e n e t w o r k c o m p o s e do f f o u r i d e n t i ca l c e l l s i n w h i c h - s y m m e t r i c H o p f b i f u r c a t i o n l e a d s t o a l l t h re e q u a d r u p e d a l g a i ts w a l k , t r o t, a n d p a c e ,w i t h t r o t a n d p a c e n o n - c o n j u g a t e . ( S e e T a b l e 4 , C o l l i n s a n d S t e w a r t [ 6 ] a n d G a m b a r y a n [ 1 3 ] fo r a d e s c ri p t i o n o ft h e s e a n d o t h e r q u a d r u p e d a l g a i t s . ) I n d e e d , s u p p o s e t h a t t h e r e is s u c h a n e t w o r k , a n d l e t F b e t h e g r o u p o f s p a t ia ls y m m e t r i e s o f t h e n e t w o r k . T h e n F c o n t a i n s Z4 a n d F i s c o n t a i n e d i n $ 4 , s o 1 /'1 = 4 , 8 , 1 2 , o r 2 4 . T o s u p p o r t t r o ta n d p a c e , F m u s t b e la r g e r t h a n 7 74 , s o I F ] > 4 .


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M. Golubitsky e t al . / Ph ysic a D 115 (1998) 56-72 61N o w I F I = 8 i m p l i e s F = 2 )4 , w h i c h c o n j u g a t e s tr o t a n d p a c e . I f I F I = 2 4 , t h e n F = 54 a n d t h e s a m e h o l d s .

S o t h e o n l y p o s s i b i l i t y i s [ F I = 1 2 . H o w e v e r , t h e r e i s n o 1 2 - e l e m e n t s u b g r o u p o f 5 4 t h a t c o n t a i n s 7 74 , b e c a u s e t h ep e r m u t a t i o n ( 1 2 3 4 ) i s o d d , s o I F I c a n n o t b e th e a l te r n a t in g g r o u p / ~ 4 , w h i c h i s t h e o n l y s u b g r o u p o f o r d e r 1 2 .

W i t h i n t h e a s s u m e d c o n t e x t , t h is g r o u p - t h e o r e t i c o b s e r v a t i o n r u l e s o u t a ll f o u r - c e ll n e t w o r k s . S i m i l a r c o n j u g a c yo b s t r u c t io n s a r i s e w h e n a t t e m p t i n g t o m o d e l 2 n - l e g g e d g a i ts w i t h 2 n i d e n ti c a l ce l ls w h e n e v e r 2 n > 4 . H o w e v e r ,f o r b i p e d a l g a i t s t h e r e is n o a n a l o g o u s a r g u m e n t t h a t r u le s o u t a t w o - c e l l n e t w o r k .

3 . T h e m o d u l a r n e t w o r k

W e n o w s h o w t h a t a n e t w o r k w i t h 4 n c e l ls a v o i d s t h e a b o v e d i f f ic u lt y , a n d a p p e a r s t o g e n e r a t e t h e a p p r o p r i a tek i n d s o f g a it s. T h e n e t w o r k i s s h o w n s c h e m a t i c a l l y i n F i g . 1 ( a) , w i t h c i r cl e s c o r r e s p o n d i n g t o t h e c o m p o n e n t c e l ls( a ll a s s u m e d i d e n t i ca l ) a n d b o n d s c o r r e s p o n d i n g t o t y p e s o f c o u p l i n g b e t w e e n c e ll s. W e e m p h a s i z e t h a t t h is f i g u rei s n o t a ' w i r i n g d i a g r a m ' , e i t h e r f o r a C P G o r f o r a n e l e c t r o n i c c i r c u i t c o n t r o l l i n g a r o b o t : i ts s o l e i n t e n t i o n i s t os h o w t h e r e q u i s it e s y m m e t r i e s i n t h e s i m p l e st m a n n e r . E a c h t y p e o f b o n d c o r r e s p o n d s t o a f ix e d t y p e o f c o u p l i n ga s s o c i a t e d w i t h a p a r t i c u la r s t r e n g t h p a r a m e t e r ( o r p a r a m e t e r s ) : a r r o w s i n d i c a te u n i d i r e c t i o n a l b o n d s a n d d o u b l el i n e s b i d i r e c t i o n a l b o n d s .

T h e s t r u ct u r e o f th i s n e t w o r k i s m o t i v a t e d b y t h e o b s e r v e d g a i t s in q u a d r u p e d s , h e x a p o d s , a n d m u l t i - l e g g e d a n i m a l ss u c h a s c e n t i p e d e s . I t r e s p e c t s t h e n a t u r a l b i l a te r a l s y m m e t r y , a n d a l s o p o s s e s s e s a 2 n - f o l d c y c l i c ' t r a n s l a t i o n a l 's y m m e t r y f r o m b a c k t o f r o n t. T h e n e t w o r k c o n s i s t s o f t w o c y c l e s o f 2 n c e l l s. F o r r e f e r e n c e p u r p o s e s t h e c e l l sa r e n u m b e r e d 1 , 3 , 5 . . . . . 4 n - 1 i n o n e c y c l e a n d 2 , 4 . . . . . 4 n i n t h e o t he r . U n i d i r e c t i o n a l c o u p l i n g s, a l l a s s u m e di d e n t i c a l i n f o r m , r u n f r o m c e l l i t o c e l l i + 2 ( r o o d 4 n ) . B i d i r e c t i o n a l c o u p l i n g s , a g a i n a s s u m e d i d e n t i c a l ( b u t

left r igh t

+ - _ U u a c(a)

front _.~@

@~/ x , , ,, . . , . , rightleft "" "~O~ ~ ~ ~k

( b )Fig. 1. (a) Schematic 4n-cell network for gaits in 2n-legged animals, on ly ceils 1 .. . . 2n are connected to legs. (b) Folding up thenetwork to eliminate long-range conn ections creates a structure with repeated modules, differing slightly at the two ends.


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62 M. Golu b i t sky e t a l. / P hys ica D 115 (1998) 56 -72differing from the unidirectional ones) run between cells 2i - I and 2i. We shall later assume that signals to thelimbs are generated only in the first half of the cells, numbers 1 . . . . 2n. This assumption introduces a necessaryasymmetry between the front and back of the animal, as well as distinguishing them f rom the section in between.Again this assumption is schematic for a wide range of alternatives with the same abstract structure: we expand onthis point below.

Note that although the diagrams have been drawn with two unidirectional couplings that are longer than theothers, this has me rely been done for convenience in representing the structure on the page. One way to make allconnections short in practice would be to fold the second part of the network, cells 2n + 1 through 4n, back over thefirst half, as illustrated in Fig. l(b). This structure is advantageous in potential applications to robotics, because itpermits a modular approach to circuit-building. While the absence of such long-range connections is also attractivebiologically, we do not wish to draw any conclusions about the layout o f animal locomo tor CPGs at this stage.

From the point of view of equivariant bifurcation theory, the most important aspect of the network is its symmetrygroup. This is

F ~ Z2n x 772. (6)The subgroup 2n cycles corresponding pairs of cells around their respective loops, and the subgroup Z2 inter-changes left and right cells in corresponding positions. Specifically, in permutation notation, ZZn is generatedby

co ---- (1 35 . . .4 n -- 1) (2 46 .. . 4n) (7)and 2~2 is generated by

~c ---- (1 2) (34 ) . . . (4n - 14 n) . (8)The group F is abelian (all elements commute) ; so there is no problem with spurious conjugac ies. In fact, we shallsee that if any periodic solution arising via Hopf bifurcation is transformed by an element of F , then the result isequivalent to a phase shift on that solution.

We now analyze the possible patterns of symmetry-breaking bifurcation that can occur in the network. Weemphasize that these patterns are model-independent in the sense that the same patterns can in principle occur inany network with the symmetry group F. This is important for applications to robotics (or CPGs) in which a morecomplex network o f connections would provide a broader range of parameters, which could simplify the practicalproblem of producing desired gait patterns as s t a b l e periodic cycles. By 'with the same symmetries' we mean thatif the schematic is modified by introducing an additional connection, then all connections that are symmetricallyrelated to it by F must also be introduced (and be identical in form). With this proviso, as many connections asdesired may be added to the schematic. For an example where this option becomes important, see Section 6.

We apply the general theory o f symmetric Ho pf (and steady-state) bifurcation to read off the patterns of symmetry-breaking that can occur in any network with symmetry grou p/~. This requires a little representation theory. Supposeas before that each cell has internal state space Ns for some s. Then the coupled network has state space

S ~ ~ 4 n s ~ i t, s X " ' " X k s (9)g

and F acts on this space by permuting the 4n compo nents (each a vector of dimension s).We first determine how the representation of F on U breaks up into real irreducible representations. As a first

step complexify the space U to get V =- C 4ns . Since F is abelian, V is the sum of one-dimensional irreducible


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M. Golubi tsky e t a l . /Phys ica D 115 (1998) 56- -72 63s u b s p a c e s ( o v e r C ) ; a n d th e s e s u b s p a c e s a r e ~ k , w h e r e 0 < j < 2 n - 1 a n d k = 0 , 1. T o d e f i n e ~ k , l e t ( = e i~r/nb e a p r i m i t iv e 2 n t h r o o t o f u n it y . T h e n r ~k i s s p a n n e d o v e r C b y

~)jk : (1 , ( - -1 )k ; i f J , ( - -1 )k ~ ' J ; f fZ j , (_ l )k ~ .2 j ; . . . ; ~ (2n -1 ) j , ( _ l ) k ~ ( 2 n - 1 ) j ) . ( 1 0 )O b s e r v e t h a t

COVjk = ~J v jk , KV jk = ( - - 1 ) k V jkSO t h a t th e s u b s p a c e l ~ k a r e i n d e e d F - i n v a r i a n t .

T h e c o r r e s p o n d i n g d e c o m p o s i t i o n o f U i n t o i r r e d u c ib l e s i s i n to t h e r e p re s e n t a ti o n s U jk w h e r e/ R e ( l ~ k ) i f j = 0 , n ,

U j k = [ R e ( V j k @ V 2 n _ j , k ) i f l < j < n - 1 . ( 11 )S e e [ 1 6 ] f o r f u r t h e r d e t a i l s.L o c a l b i f u r ca t i o n s f r o m t h e t r i v i a / s o l u t io n o c c u r w h e n t h e l i n e a r iz a t i o n ( J a c o b i a n m a t r i x ) h a s z e r o e i g e n v a l u es

( s t e a d y - st a t e b i f u r c a ti o n ) o r p u r e l y i m a g i n a r y e i g e n v a l u e s ( H o p f b i f u r ca t i o n ). G e n e r i c a ll y , t h e z e r o e i g e n s p a c e i sa n a b s o l u t e l y i r r ed u c i b l e r e p r e s e n t a t io n f o r s t e a d y - st a t e b i f u rc a t i o n , a n d i s / " - s i m p l e f o r H o p f b i fu r c a t i o n ( e i t he r ad i r e c t s u m o f t w o i s o m o r p h i c c o p i e s o f a n a b s o l u t e l y i r r e d u c i b le r e p r e s e n t a t io n o r a s i n g l e n o n a b s o l u t e l y i r r e d u c i b l er e p r e s en t a t io n ) . T h a t i s , th e n e t w o r k e q u a t i o n s c a n u n d e r g o s t e a d y -s t a te b i f u r c a t i o n s c o r r e s p o n d i n g t o t h e s u b s p a c e sUo o , Un o , U01, U n l . T h e n e t w o r k c a n u n d e r g o H o p f b i f u r ca t io n c o r r e s p o n d in g t o a n y o f t h e U jk w h e n s > 1 , b u tw h e n s = 1 t h e o n l y H o p f b i f u r c a t io n s f r o m t h e tr i vi a l b r a n c h a r e t h o s e c o r r e s p o n d i n g t o e i g e n s p a c e s f o r w h i c hj = 1 . . . . n - 1 . ( T h e r e a s o n f o r t h is r e s tr i c ti o n is t h e c o n d i t i o n o f / " - s i m p l i c i t y : i m a g i n a r y e ig e n v a l u e s c a n n o to c c u r u n l e s s t h e r e p r e s e n t a t i o n s o f r e a l t y p e h a v e m u l t i p l i c i t y a t l e as t 2 . ) O u r d i s c u s s i o n i n S e c t i o n 6 t r e a ts t h e c a s es = 2 . ( S t a n d a r d m o d e l e q u a t i o n s f o r n e u r o n s h a v e s > 2 , s e e [3 ] . )

T h e e f f e c t o f t h e a s s u m e d b i l at e ra l s y m m e t r y o f th e n e t w o r k ( t h e x s y m m e t r y ) c a n b e d e t ec t e d i m m e d i a t e l y . H o p fb i f u r c a t io n s c o r r e s p o n d i n g t o th e s u b s p a c e U jk l e a d t o g a i ts i n w h i c h 0 c s y m m e t r i c ) p a i rs o f le g s a r e i n p h a s e w h e nk = 0 a n d h a l f a p e r i o d o u t o f p h a s e w h e n k = 1 .

4 . P r i m a r y b i f u r c a t io n s4 . 1 . B i p e d a l g a i t s

R e c a l l i n g t h a t n i s t h e n u m b e r o f p a i r s o f l e g s w e s e e th a t t h e c a s e 2 n = 2 r e f e rs t o b i p e d a l g a i t s. T h i s c a s e i se x c e p t i o n a l b e c a u s e o f i ts s i m p l i c it y . F o r o u r p r o p o s e d f o u r - c e l l n e t w o r k t h e a n a l y s is o f b ip e d a l g a i t s l e a d s t o t h es a m e p a t t e rn s a s a t w o - c e l l n e t w o r k [ 6 ]: a f te r a p r i m a r y H o p f b i f u r c a ti o n t h e l e g s a r e e i t h er i n p h a s e o r h a l f a p e r i o do u t o f p h a s e.

W e r e m a r k t h a t o t h e r p h a s e r e la t i o n sh i p s , s u c h a s s e e n i n t h e h u m a n g a i t ' s k i p ' , c a n o c c u r v i a s e c o n d a r y p e r io d -d o u b l i n g b i f u r c a t i o n s . R e c a l l t h a t i n s u c h a b i f u r c a t i o n a p e r i o d i c s t a te x ( t ) o f p e r i o d T m a k e s a tr a n s i t i o n t o a s t a teo f p e r i o d 2 T , i n e f f e c t b y m o d i f y i n g t w o s u c c e s s iv e p e r i o d s o f x ( t) i n t w o s l ig h t l y d i ff e r e n t w a y s . S u p p o s e t h a t ab i l a te r a l ly s y m m e t r i c s y s t e m h a s a T - p e r i o d i c s ta t e ( Xl ( t ), x 2 ( t ) ) , w h e r e t h e s y m m e t r y p e r m u t e s t h e i n d ic e s 1 a n d2 . T h e n i t is p o s s i b l e f o r s u c h a s t a t e t o u n d e r g o a p e r i o d - d o u b l i n g b i f u r c a t i o n t o a s ta t e ( Y l ( t ) , y 2 ( t ) ) t h a t h a s as p a t i o - t e m p o r a l s y m m e t r y :

y2(t) -----y l ( t + T ) .


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64 M. Golubitsky et al./Physica D 115 (1998) 56-72T h i s s y m m e t r y i s o b s e r v e d i n ( o n e f o r m o f ) th e s k i p, a n d i s a ls o i n t h e ' h a l f - b o u n d ' o f s m a l l a n i m a l s s u c h a s t h ej e r b o a , s e e [ 1 3 ] .4.2 . Q uadru pedal ga i t s

T h e d i f f e r e n c e b e t w e e n o u r 4 n - c e l l m o d e l a n d 2 n - c e l l m o d e l s is e v i d e n t w h e n 2 n = 4 , t h e c a se o f q u a d r n p e d s : i nt h is s e c t i o n w e f o c u s o n t h a t c a s e. E a c h c o m p l e x i r r e d u c i b le r e p r e s e n t a t i o n o f F s u p p o r t s e i th e r o n e o r t w o p r i m a r yb i f u r c a t io n s t o p h a s e - l o c k e d p a t te r n s o f o s c il l at i o n , i n w h i c h t h e p h a s e s a r e d e t e r m i n e d b y t h e r o o t s o f u n i t y i n ( 1 0 ) .S p e c i f i c a l l y , o n t h e s p a c e Ujk t h e p h a s e s h i f t o n c e l l Cm i s :

z~(m - 1 ) j i f m i s odd ,~ t z (12)

( m - 2 ) + ~ i f m i s e v e n ,w h e r e e i t h e r t h e + o r - s i g n i s c h o s e n t h r o u g h o u t , a n d f o r j = 0 o r j = n o n l y t h e + s i g n i s u s e d . ( T h e r e a s o n f o rt h i s r e s tr i c t i o n i s t h a t f o r p h a s e s o f 0 o r I a s i g n c h a n g e h a s n o e f f e c t .)

W e m a y n o w r e a d o f f t h e c o r r e s p o n d i n g p a t t e r n s f o r t h e p r i m a r y b i f u r c at i o n s. W e d e n o t e t h e m b y a n a r ra y o fp h a s e s h if ts c o r r e s p o n d i n g t o c e ll s a r r a n g e d i n t h e f o r m7 85 63 41 2

R e c a l l t h a t f o r p u r p o s e s o f i n t e r p r e t a ti o n , w e a s s u m e t h a t o n l y c e l ls 1 - 4 d r i v e t h e l eg s , s o t h a t :- c e l l 1 d r ive s the l e f t r e a r l eg ,- c e l l 2 d r i v e s t h e r i g h t r e a r l e g ,- c e l l 3 d r i v e s t h e le f t f r o n t l e g ,- c e l l 4 d r i v e s t h e r i g h t f r o n t l e g .T h u s t h e p h a s e s o f c e ll s 5 - 8 a r e i g n o r e d a s f a r a s t h e g a i t p a t t e r n s a r e c o n c e r n e d : t h o s e c e l l s a r e p r e s e n t i n o r d e rt o p r o p a g a t e t h e d y n a m i c s c o r r e c tl y . A n a l te r n a ti v e a s s u m p t i o n w i t h t h e s a m e c o n s e q u e n c e s f o r g a i t p a t t e rn s i s th a tt h e l e f t r e a r l e g i s d r i v e n b y s o m e a sy m m e t r i c f u n c t i o n o f t h e s t a te s o f c e l ls 1 a n d 5 , a n d t h e s a m e f u n c t i o n o f c e l ls2 a n d 6 , 3 a n d 7 , a n d 4 a n d 8 i s u s e d t o d r i v e t h e r i g h t re a r , l e ft f ro n t , a n d r i g h t f r o n t l e g s, r e s p e c t i v e ly . T h e ' w i r i n gd i a g r a m ' f o r s u c h a c h o i c e i n t r o d u c e s l o n g - r a n g e c o n n e c t i o n s i n a c o n t r o l n e t w o r k f o r m u l t i - l e g g e d l o c o m o t i o n -t h is i s n o t a n o b s t a c l e i n r o b o t i c a p p l i c a ti o n s , w h e r e l o n g - r a n g e c o n n e c t i o n s c a n b e m a d e e f f e c ti v e l y as r a p id a ss h o r t - r a n g e o n e s .

T h e p r i m a r y p a t t e r n s , i n t h i s n o t a t i o n , a r e f o u n d i n T a b l e 1 .4.3 . He xapo dal ga i t s

W h e n 2 n = 6 w e h a v e a 1 2 - c e l l n e t w o r k , a n d i n th e s a m e n o t a t i o n t h e p r i m a r y H o p f b i f u r c a t io n s f r o m t h e t ri v ia ls t a te a r e l i s t e d i n T a b l e 2 . F r o m t h a t t a b l e , w e s e e t h a t g a i t s U 0 0 a n d U 01 a r e th e i n s e c t a n a l o g o f p r o n k a n d p a c e , a n dg a i t s U 31 a n d U l l a r e t h e s t a n d a r d t r i p o d a n d m e ta c h ro n a l g a it s . ( S e e [ 7 ] f o r a d e s c r i p t io n o f c o m m o n h e x a p o d a lga i t s . )

G a i t U 30 i s a s o m e w h a t c u r i o u s g a i t w h i c h w e c a l l t h e lurch. I n g a i t U 1 0 t h e a n i m a l ( o r r o b o t ) m o v e s l e g s o n l e f ta n d r i g h t i n s y n c h r o n y i n a w a v e t h a t p a s se s f r o m b a c k t o f r o n t, a n d t h e n a p p e a r s t o p a u s e b e f o r e c o n t i n u i n g . T h i s


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M. Golubitsky et al./Physica D 115 (1998) 56-72Table 1Quadrupedal gai ts ( in jum p and wa lk choose ei ther 4- or - throughout)

65

Uoo Uoa U~o UH U2o U2~] i ] 1 00 0 : 4.43 =E3 3 q_ l 2 2 :1 1 1 1 10 0 0 ~ ~ ~ ~ 0 0 0 0

1 1 1 00 o o 2 21 1 10 0 0 ~ 0 0 0 ~ 0 0 0

Pron k Pace Jump =k W alk t : Bound TrotTable 2Hexapodal g ai ts (choose + or -- throughout in gai ts where relevant)UO0 UO1 U30 U31 UIO U ll U20 U21

1 i 1 i0 0 0 ~ 2 2o o o o

i l 10 0 0 2 2 ~ 0o o o o o

1 1 10 0 0 i ~ ~ ~ 0o o o o oPronk Pace Lurch Tripod

0 : t=5 5 =t5

1 1 1 0 0 0 0 1

1 10 0 0 ~ 0 0 0Inchworm ~: Metachronal -~ Caterp i l lar+ Roll ing tr ipod

w e c a l l th e i n c h w o rm g a i t b e c a u s e i t i s r e m i n i s c e n t o f t h e m o t i o n o f t h a t a n i m a l . G a i t U 2 0 i s a n o t h e r g a i t in w h i c ht h e m o t i o n o f l e f t a n d r i g h t l e g s a r e i n s y n c h r o n y . W e c a l l it t h e caterp i l lar g a i t .

F i n a l l y , in g a i t U 2 1 tw o w a v e s t r a v e l f r o m b a c k t o f ro n t , o n e t h i r d o f a p e r i o d o u t o f p h a s e , b u t l e f t a n d r i g h t s i d e sa r e h a l f a p e r i o d o u t o f p h a s e . S o t h e o r d e r i n w h i c h t h e l e g s m o v e i s b a c k l e f t, f r o n t r i g h t , m i d d l e l e f t, b a c k r i g h t ,f r o n t r ig h t , m i d d l e r i g h t - a t e q u a l l y s p a c e d i n t e r v a l . T h i s i s a k i n d o f roll ing tripod.4 . 4. M a n y - l e g g e d g a i t s

A s i m i l a r a n a l y s i s a p p l ie s to n e t w o r k s o f t h e k i n d t h a t m i g h t b e a s s o c i a t e d w i t h m a n y - l e g g e d a n i m a l s s u c h a sc e n t i p e d e s a n d m i l l ip e d e s - a n d a l s o l e g le s s a n i m a l s s u c h a s e e l s, s n a k e s , a n d l a m p r e y , i n w h i c h ' l e g ' s h o u l d b ei n t e r p r e t e d a s ' m u s c u l a r u n i t '. T h e r a n g e o f g a i t p a t t e rn s i s c o n s i d e r a b l e , b u t t h e y a l l h a v e t h e s a m e f u n d a m e n t a lc h a r a c t e r i s t i c s F i r s t , s e l e c t s o m e d i v i s o r d o f 2 n , t h e n u m b e r o f c e l l s o n o n e s i d e o f t h e n e t w o r k , b u t a l s o e q u a l t ot h e t o t a l n u m b e r o f l e g s . T h e n g r o u p t h e c e l l s o n th e l e f t s i d e i n to 2 n / d c l u m p s , e a c h s p a c e d d c e l l s a p a r t . T h e p h a s er e l a t io n s h i p s a l o n g t h a t s id e o f t h e n e t w o r k a r e l ik e t h o s e o f a ( 4 n / d ) - c e l l n e t w o r k o f th e s a m e k i n d , b u t r e p e a t e dd t i m e s i n t h e ' f o r w a r d ' d i r e c t i o n . T h e l e f t - r i g h t p a i r s o f c e l l s e i t h e r a r e i n p h a s e t h r o u g h o u t t h e c y c l e , o r h a l f ap e r i o d o u t o f p h a s e t h r o u g h o u t t h e c y c l e . In c e n t i p e d e s , m i l li p e d e s , a n d l a m p r e y , t h e ' h a l f a p e r i o d o u t o f o p t i o n 'o p t i o n i s t h e o n e o b s e r v e d , s o w e f o c u s o n t h a t e x a m p l e .

S i m i l a r l y c l u m p e d t r a v e l i n g w a v e s c a n b e p r o d u c e d i n l i n e a r c h a in s o f c e ll s b y f o r c i n g t h e m a t t h e e n d a n di m p o s i n g s u i ta b l e b o u n d a r y c o n d i t i o n s a t th e f r e e f ro n t e n d ( th e m a t h e m a t i c a l l y m o s t t r a c ta b l e b e i n g p e r i o d i cb o u n d a r y c o n d i t io n s ) . T h e s a m e g o e s f o r p a i rs o f c h a in s , c r o s s - c o n n e c t e d t o i m p o s e a h a l f - p e r i o d p h a s e s h if t. Am o d e l o f th i s s o r t o c c u r s i n t h e w o r k o f R a n d e t a l. [ 2 5] . I n s u c h a s y s t e m t h e r e i s n o p r e f e r r e d ' w a v e n u m b e r ' f o rt h e t r a v e l i n g w a v e s : t h e y c a n b e a n y i n t e g e r , a n d a r e n o t r e l a t e d t o t h e n u m b e r o f l e g s ( o r m u s c u l a r u n i ts i n c a s e ss u c h a s t h e l a m p r e y ) .


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66 M . G o l u b i t s ky e t a l . / P h y s i c a D 1 1 5 ( 1 9 98 ) 5 6 - 7 2

" J l~ waYe w ~ v e Iw a Y e

Fig. 2. Figure fr om [1]. Thick lines indicate legs in contact with the ground. (L eft) The centipede has 3 waves. (Right) The cen tipede has3/2 waves.

T h i s f e a t u r e l e a d s t o a p r e d i c ti o n t h a t i s s p e ci f ic t o s y m m e t r y - b a s e d m o d e l s o f t h e t y p e w e a r e c o n s i d er i n g .N a m e l y , t h a t t h e w a v e n u m b e r ( t h e s p a ti a l p e r i o d b e t w e e n s y n c h r o n o u s l y m o v i n g l e g s, m e a s u r e d a l o n g t h e l e n g t ho f t h e a n i m a l ) s h o u l d a l w a y s d i v i d e t h e t o t a l n u m b e r o f l eg s ( o r m u s c u l a r u n i t s i n c a s e s s u c h a s th e l a m p r e y ) . F o re x a m p l e i n a 3 0 - l e g g e d c e n t i p e d e ( n = 1 5 ) th e p r e d i c t e d w a v e n u m b e r s a r e 1 , 2 , 3 , 5 , 6 , 10 , 1 5 . I n a l i n e a r c h a i nm o d e l , a ll n u m b e r s f r o m 1 t o 1 5 a r e p o s s ib l e ; i n a c y c l i c n e t w o r k m o d e l w i t h 2 n c e l l s i n st e a d o f 4 n , o n l y d i v i s o r so f n s h o u l d a p p e a r - i n t h is c a s e 1 , 3 , 5 , 1 5 , e x c l u d i n g 2 , 6 , 1 0 .

A n o t h e r w a y s t o s t a te t h is , w h i c h i s l es s s e n s it i ve t o i m p e r f e c t i o n s i n t h e n e t w o r k , i s th a t t h e n u m b e r o f w a v e st h a t f it s i n t o t h e ' d i r e c t l y o b s e r v a b l e ' h a l f o f t h e n e t w o r k , c e l l 1 t h r o u g h 2 n , i s e i t h e r a n i n t e g e r o r h a l f - a n - o d di n te g e r. T h e h a l f - a n - o d d i n t e g e r p a tt e r n s a r e c h a r e c te r i st i c o f ' d o u b l e d ' n e t w o r k s s u c h a s t h a t d e s c r ib e d i n t h is p a p e r( s ee [ 1 1 ]) , a n d s e r v e a s a n i n d i r e ct i n d i c a t o r o f th e p r e s e n c e o f t h e h i d d e n h a l f o f t h e n e t w o r k . I n t h e c o n t e x t o fp o s s i b l e a p p l i c a t io n s t o C P G s , i t m a y t h e r e f o r e b e s i g n if i c an t t h a t A l e x a n d e r [ 1 ] p r o v id e s d r a w i n g s o f s e v er a l g a it sb y a 4 0 - l e g g e d c e n t i p e d e . I n a l l c a s e s t h e n u m b e r o f w a v e s a p p e a r s t o b e c l o s e e i t h e r t o a n i n t e g e r ( 4 , 3 , 2 ) o rh a l f - a n - o d d i n t e g e r ( 3 / 2 ) . I n p a rt i cu l a r, t h e h a l f - a n - o d d i n t e g e r p a t t e r n d o e s s e e m t o o c c u r. S e e F i g . 2 . T h e n u m b e ro f l e g s d e p i c t e d i n t h e s p a t i a l p e r i o d o f t h e v a r i o u s t r a v e l i n g w a v e s a r e n o t a l l e x a c t d i v i s o r s o f 8 0 : f o r e x a m p l eo n e p i c t u r e s h o w s 1 3 le g s p e r w a v e . H o w e v e r , t h e y a r e v e r y c l o s e t o e x a c t d i v is o r s. F o r e x a m p l e 1 3 i s v e r y c l o s et o 8 0 / 6 = 2 0 x 2 / 3 , w h e r e 2 0 is t h e n u m b e r o f l e g s a lo n g o n e s i d e o f t h e a n im a l . S o h e r e th e n u m b e r o f w a v e sp a s s i n g a l o n g t h e a n i m a l i s v e r y c l o s e t o 3 / 2 .

P r e c i s e n u m e r i c a l a g r e e m e n t w o u l d b e u n l i k e l y f o r s e v e r a l r e a s o n s : f ir st , th e s y m m e t r y o f a re a l C P G w i l l b eo n l y a p p r o x i m a t e ; a n d s e c o n d , a n a n i m a l C P G , e v e n i f i t i s o f t h e f o r m w e a r e s t u d y i n g h e r e , m i g h t i n v o l v em o r e c e l l s th a n t h e r e a r e l e g s ( f o r i n s t a n ce b y a s s i g n i n g a c h a i n o f m o d u l e s t o e a c h l e g , w h i c h i n t h is c a s e w o u l dl e a d t o a m u l t i p l e o f 8 0 c e l ls ) . T h e p r e d i c t i o n t h a t h a l f - a n - o d d i n t e g e r w a v e s c a n o c c u r is r o b u s t u n d e r s u c hc h a n g e s .

T h e w a v e l i k e m o t i o n o f t h e l a m p r e y s h o w n i n [ 2 0 ] s h o w s 5 / 2 w a v e s a l o n g t h e a n i m a l ' s l e n g t h , b o t h i n f o r w a r da n d b a c k w a r d m o t i o n : t h i s i s e s p e c i a l l y c l e a r i n t h e t h i r d a n d f i f t h i m a g e s i n t h e t o p f i g u r e o n p a g e 5 1 . A g a i n t h i sr e l a t io n s h i p i s c o n s is t e n t w i t h o u r m o d e l b u t r e q u i r e s s p e c ia l e x p l a n a t io n f o r l i n e ar c h a i n m o d e l s .


12/17

x 3 ( t ) = x l ( t -k- )x s ( t ) = x l ( t - F 1 ) a n dx 7 ( t ) = x l ( t + 3 ) a n d

T h e (K , ) s y m m e t r y f o r c e sx2 ( t ) = Xl ( t "q- ) .

M. Golubitsky et al. /Physica D 115 (1998) 56--72 675 . S e c o n d a r y q u a d r u p e d a l g a it s

A s d i s c u s s e d p r e v i o u s l y , p e r i o d i c s o l u t io n s x ( t ) h a v e s p a t i o - t e m p o r a l s y m m e t r i e s w h i c h f o r m a s u b g r o u p Z CF x N 1 . S o ( or , 0 ) i s a s y m m e t r y o f x ( t ) i f

c r x ( t + O ) = x ( t ) , ( 1 3 )a n d E c o n s i s t s o f a ll p a ir s ( o -, 0 ) s a t i s f y i n g ( 1 3 ) . I n o u r m o d u l a r n e t w o r k f o r q u a d r u p e d s

x ( t ) - ~ ( X l ( t) , x 2 ( t ) . . . . . x 8 ( t ) ) ,w h e r e x j ( t ) d e s c r i b e s t h e s i g n a l i n c e l l j o f t h e C P G . W e t h e n i d e n t i fi e d

( x l ( t ) , x 2 ( t ) , x 3 ( t ) , x 4 ( t ) )a s t h e s i g n a ls f r o m t h e C P G t h a t d r iv e t h e f o u r l eg s o f t h e q u a d r u p e d .

I n S e c t i o n 4 .2 w e l i s te d th e p r i m a r y g a i t s g e n e r a t e d b y t h e F = Z 4 Z 2 s y m m e t r y o f o u r n e t w o r k . T h e s e g a i tss a t i s fy t w o p r o p e r t i e s:( a ) T h e s ig n a l s x j ( t ) a ll h a v e i d e n t i c a l w a v e f o r m s ( f o r c e d b y s y m m e t r y ) a n d d i f f e r o n l y b y p h a s e s h i f ts , a n d( b ) P e r i o d i c s o l u t i o n s w i t h t h e s e s y m m e t r i e s c a n b e e x p e c t e d t o a p p e a r in m o d e l e q u a t i o n s b y H o p f b i f u r c a t i o n

f r o m t h e t ri v i a l s t e a d y - s t a t e ' s t a n d ' .F o r s i m p l i c it y w e a s s u m e x ( t ) h a s p e r i o d 1 . A s a n e x a m p l e , r e c o n s i d e r t h e w a l k . I n o u r n e t w o r k , t h e s y m m e t r y

g r o u p o f a w a l k i s g e n e r a t e d b y( co , l ) a n d (K, 1 ) ,

w h e r ec o = ( 1 3 5 7 ) ( 2 4 6 8 ) a n d K ~ ( 1 2 ) (3 4 ) (5 6 ) (7 8 ).

T h e ( if , ) s y m m e t r y f o r c e s th e c o m p o n e n t s x j ( t ) t o s a t i s f ya n d x 4 ( t ) = x 2 ( t + ) ,

x 6 ( t ) = x 2 ( t q - 1 ) ,x8( t ) ~ -- -y 2 ( t q - 3 ) .

C o n s e q u e n t l y , t h e f o u r s i g n a l s s e n t b y t h i s C P G a r e( X l ( t ) , x 2 ( t ) , x 3 ( t ) , x 4 ( t ) ) = ( X l ( t ) , X l ( t q - 1 ) , x l ( t q - 1 ) , X l ( t q - 3 ) ) ,

t h a t is , t h e s a m e w a v e f o r m ( x l ( t ) ) i s s e n t to e a c h l e g - b u t w i t h t h e c h a r a c t e r is t i c p h a s e s h i ft s a s s o c i a t e d w i t h a' w a l k ' .I n th i s s e c t i o n w e d i s c u s s a l l o f th e s y m m e t r i e s a s s o c i a t e d t o p e r i o d i c s o l u t io n s i n o u r m o d u l a r n e t w o r k . B y

m a k i n g t h is c l a s s i f ic a t i o n w e w i l l f in d s e c o n d a r y g a i t s t h a t m a y b e a s s o c i a t e d t o r o t a r y g a l l o p s , t r a n s v e r s e g a l l o p s ,a n d c a n t e r s , a s w e l l a s o t h e r p o s s i b i l i t i e s . B y t h e t e r m s e c o n d a r y g a i t w e m e a n t h a t t he C P G s e n d s t w o d i ff e re n ts i g n a l s t o t w o p a i r s o f l e g s . N e v e r t h l e s s , i t i s p o s s i b l e t o a s s o c i a t e a p a t t e r n o f p h a s e s h i f t s f o r t h e s e g a i t s , a s w ee x p l a i n b e l o w .


13/17

6 8T a b l e 3L a t t ic e o f s u b g r o u p s o f F = ~ 4 J ~2

M. Golubitslcy et al./Physica D 115 (1998) 56-72

2~4((0 ~2(tC)/ * \~4 (0 ) ) D2(K , 0 )2 )

\ 1" / 1" \2Z-2 0)2 ) ~2 (g ) 772(K0) )\ "t /"1

5.1. Classif ication of isotropy subgroup sT h e s t r a t eg y f o r c la s s i f y i n g a ll i s o t r o p y s u b g r o u p s o f p e r i o d i c s o l u t io n s w o r k s a s f o l l o w s . L e t K = I ? M F b e

t h e s u b g r o u p o f space s y m m e t r i e s t h a t f i x e a c h p e r i o d i c s o l u t i o n p o i n t w i s e , t h a t i s k c K i f kx ( t ) = x ( t ) f o r a l lt i m e t . F o r e x a m p l e , K is a s p a c e s y m m e t r y o f a n y g a i t w h e r e l e f t a n d r i g h t l e g s m o v e i n u n is o n , s u c h a s th e p r o n ko r b o u n d . L e t K C F b e t h e s u b g r o u p o b t a i n e d f r o m E b y s t r i p p i n g o f f t h e p h a s e s h i ft , t h a t i s, if (( 7, 0 ) E 2 ; , t h e nc r E L . I t f o l l o w s th a t K i s a s u b g r o u p o f H a n d H / K i s a fi n it e s u b g r o u p o f 5 1. H e n c e , H / K i s c y c l i c . O u rs t r a t e g y d i v i d e s i n t o t h r e e p a r t s :1 . C la s s i fy a l l s ub gro up s L o f F = 774 772.2 . F o r e a c h L c l a s s if y a ll n o r m a l s u b g r o u p s K s u c h th a t H / K i s cyc l i c .3 . F o r e a c h o f t h e a d m i s s i b le p a i r s H / K d e t e r m i n e t h e c o n s t r a i n t s o n t h e si g n a l x ( t ) t o d e c i d e w h e t h e r t h e a s s o c i a t e d

g a i t is s e c o n d a r y .T h e l a t t i c e o f s u b g r o u p s o f 2 4 x 772 i s g i v e n i n T a b l e 3 w h e r e c o i s t h e g e n e r a t o r o f 7/4 a n d t c i s t h e g e n e r a t o r o f

7/2.T h e p o s s i b l e c h o i c e s f o r H a n d K a r e li s t e d i n T a b l e 4 . I n th i s t a b l e w e i n d i c a t e t h e re s t r i c ti o n s o n t h e s ig n a l s

f r o m t h e n e t w o r k g o i n g t o e a c h l e g . W e a s s u m e t h a t X l ( t ) i s a g i v e n w a v e f o r m a n d w r i t e, w h e r e a p p r o p r ia t e , t h er e s t r i c t io n s o f x 2 ( t ) , x 3 ( t ) , a n d x 4 ( t ) . W e d o n o t i n d i c a t e t h e r e s t r i c t i o n s o n x 5 ( t ) . . . . . x 8 ( t ) .

W e d e f i n e t h e twist type t o b e t h e o r d e r o f t h e c y c l i c g r o u p H / K . T h e g r o u p e l e m e n t s i n H - K c o r r e s p o n d t os p a f i o - t e m p o r a l s y m m e t r i e s. W h e n t h e t w i s t ty p e i s 2 , th e o n l y p h a s e s h if t is . W h e n t h e t w i s t t y p e is 4 , th e r ea r e t w o t y p e s o f p h a s e s h i ft p o s si b l e. T h e g e n e r a t o r o f H / K c a n b e p h a s e s h i f t e d b y 1 o r - . W e i n d i c a te t h e s ep o s s i b i l i ti e s b y 4 + a n d 4 - .

T h e r e a r e e i g h t p r i m a r y g a i ts - a s w e h a v e d i s c u s s e d p r e v i o u sl y . T h e s e c o n d a r y g a i t s a r e t h o s e t h a t h a v e t w oi n d e p e n d e n t w a v e f o r m a n d w e h a v e i n d i c a t e d t h e 1 4 s e c o n d a r y g a i t s li s te d i n t h is t a b l e b y a * .

5.2. Secondary gai tsN e x t w e d i s c u s s t h e g a i t p a t t e r n t h a t is a s s o c i a t ed w i t h t h e s e c o n d a r y g a i ts o f t h e p r e v i o u s s u b s e c ti o n . W e a s s u m e

t h a t t h e s i g n a l s x j ( t ) p r o d u c e a p p r o x i m a t e l y s i n u s o i d a l m o v e m e n t i n th e l eg s , th a t is , w e a s s u m e f o r t h e p u r p o s e o fi d e n t i f y i n g g a i t s t h a t

x j ( t ) ~ A j s in (2J r ( t + qg j ) )f o r s o m e a m p l i t u d e A j a n d s o m e p h a s e s h i f t ~0 j.

F o r e x a m p l e , c o n s i d e r t h e s e c o n d a r y g a i t s t h a t w e h a v e c a l l e d r o t a r y g a l lo p , t r a n sv e r s e g a l l o p a n d c a n t e r. A s s u m ea f t er n o r m a l i z a t i o n o f t h e p h a s e s h i ft a n d a m p l i t u d e o f x ] ( t ) t h a t

X l ( t ) ~ s i n ( 2 z r t ) a n d X 2 ( t ) ~ A 2 s i n ( 2 z r ( t + q ) 2 ) ) .


14/17

M . G o l u bi ts k y e t a l . / P h y s i c a D 1 1 5 ( 1 9 9 8 ) 5 6 - 7 2 6 9T a b l e 4P a i r s o f H a n d K f o r q u a d r u p e d sH K N a m e T w i s t x 2 ( t ) x 3 ( t ) x 4 ( t )2 ~ 4 (0 ) 2 ~ 2( K 2 ~ 4( 0 ) x Z 2 ( t c ) P r o n k 0 X l ( t ) X l ( t ) X l ( t )

7 7 4 (0 ) R a c k 2 X l ( t + 1 ) X l ( t ) X l ( t + 1 )D 2 0 c , 0 )2 ) B o u n d 2 x l ( t ) X l ( t + 1 ) x a ( t + 1 )7 7 4 (t c0 ) T r o t 2 X l ( t + 1 ) X l ( t + 1 ) X l ( t )7 72 (K J u m p 4 + x l ( t ) x i ( t + 1 ) X l ( t + 1 )

4 - x l ( t ) x l ( t - I ) X l ( t - - I ) 2 (K O ) 2 ) W a l k 4 + x l ( t q - 1 ) x l ( t + ) X l ( t + 3)4

4 - x l ( t - I ) X l ( t - i ) x l ( t - I ))7 4 ( I t 0 ) ) Z 4 (x 0 ) ) * Lo p i n g t ro t 0 x 2 ( t ) x 2 ( t ) x 1 ( t )

7 / 2 (0 ) 2 ) * R o t a r y g a l l o p 2 x 2 ( t ) x 2 ( t + 1 ) X l ( t + 1 )1 " R o t a r y c a n t e r 4 + x 2 ( t ) x 2 ( t + 1 ) X l ( t + 1 )

4 - x i ( t ) x 2 ( t _ 1 ) Xl ( t __ I )~ 4 ( 0 )) ~ 4 ( 0 )) * L o p i n g r a c k 0 x 2 ( t ) x 1 ( t ) x 2 ( t )

7 7 2 (o ) 2) * T r a n s v e r s e g a l l o p 2 x z ( t ) x l ( t + 1 ) x 2 ( t + 1 )1 " C a n t e r 4 + x z ( t ) X l ( t + 1 ) x z ( t + 1 ))4 - x 2 ( t ) x l ( t _ 1 ) x 2 ( t - - 4

D Z ( K 0 ) 2 ) D Z ( X 0 ) 2 ) * L o p i n g b o u n d 0 x 1 ( t ) x 3 ( t ) x 3 ( t )7 /2 ( 0 ) 2 ) * R u n n i n g w a l k 2 x 1 ( t + 21-) x 3 ( t ) x 3 ( t + ~2 (KOO2)* R u n n i n g w a l k 2 x 1 ( t + 1 ) x 3 ( t ) x 3 ( t + 7 / 2 0 c )* L o p i n g b o u n d 2 x 1 ( t ) x 3 ( t ) x 3 ( t )

7 / 2 (0 9 2 ) 7 / 2 (0 )2 ) 0 X2 ( t ) X3 ( t ) X ( t )1 2 x 2 ( t ) x 3 ( t ) X 4 ( t )

7 / 2 ( g ) Z 2 ( K ) * L o p i n g b o u n d 0 X l ( t ) x 3 ( t ) x 3 ( t )1 " R u n n i n g w a l k 2 X l ( t + ) x 3 ( t ) x 3 ( t + 1 )~ 2 ( K 0 ) 2 ) ~ 2 (K (.O ) 0 x 2 ( t ) x 3 ( t ) x 4 ( t )

1 2 x 2 ( t ) x 3 ( t ) x 4 ( t )1 1 0 x 2 ( t ) x 3 ( t ) x 4 ( t )

The ampl i t ude A 2 d oes no t af fec t t he det erminat ion of t he t ime w h en a g iven leg s tr ikes t he ground . A ssu min g t hatt hese approxim at ions are accurate , w e can use t he in f ormat ion in Tab le 4 t o descr ibe t he order t hat t he l egs s t riket he ground and t he ir re lat ive phase . T he s ignals t o t he l egs are:

x2 ( t + l ) X l ( t + l )X l ( t ) X 2 ( t )ro ta ry ga l lop

This l eads to re l a t ive phases :1 1~ o + ~

0 ~orotary

10

gal lop t ransverse

X l ( t q - 1 ) x 2 ( t q - 1 )X l ( t ) X 2 ( t )t ransverse ga l lop

q)ga l lop

x l ( t + 1) x : ( t + )Xl( t ) x2 ( t )can ter

1 10 q)

can ter


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70 M. Golubi tsky e t a l . /Phys ica D 115 (1998) 56-72Table 5Parameter values where different gai ts have been observedGa it c ~ f l VPronk 0.5 0.01 0.014 0.025 0.02Pace 0 .44 0 .025 0 .02 -0 . 01 -0 . 01 2Bound 0 .44 -0 . 01 -0 .0 10 2 0 .025 0 .02Tr ot 0 .44 - 0 . 02 - 0 .00 2 - 0 .0 25 0 .015Jump 0.44 -0 .0 2 0 .01 0 .025 0.015Walk 0 .44 -0 . 01 0 .0102 -0:0 25 0 .02

W h e n ~p i s s m a l l i n t h e f ir s t t w o g a i t s w e o b t a i n t h e s t a n d a r d p h a s e d e s c r i p t i o n o f t h e r o t a r y g a l l o p a n d t h e t r a n s v e r s eg a l l o p . I n th e t h i r d c a s e , w h e n ~0 ~ , w e h a v e t h e s t a n d a r d p h a s e d e s c r i p t i o n o f t h e c a n t e r . T h e r e a r e s e v e r a l o t h e rp h a s e r e l a t i o n s p o s s i b l e fr o m t h e o t h e r s e c o n d a r y g a i t s , b u t w e d o n o t e n u m e r a t e t h e m h e r e .

6 . N u m e r i c a l r es u l ts f ro m m o d e l e q u a t i o n sI n t h i s s e c t i o n w e c o n s i d e r t h e e x i s t e n c e o f t h e p r i m a r y g a i t s i n a s p e c i fi c s y s t e m o f d i f f e r e n t i a l e q u a t i o n s . W e a r e

p a r t i c u l a r l y i n t e r es t e d i n s t a b l e p e r i o d i c s o l u t io n s t h a t a r e o b t a i n e d i n o u r n e t w o r k v i a H o p f b i f u r c a ti o n . T o s p e c i f yt h e n e t w o r k e q u a t i o n s , w e n e e d t o c h o o s e t h e d i f f e r e n t ia l e q u a t i o n s m o d e l i n g t h e i n t e r n a l d y n a m i c s a n d t h e p r e c i s ef o r m o f c o u p l in g .

I n o r d e r t o h a v e a l l H o p f b i f u r c a ti o n s p r e s e n t i n th e n e t w o r k e q u a t i o n s , t h e i n t e rn a l d y n a m i c s m u s t b e a t l e a s tt w o - d i m e n s i o n a l . W e a s s u m e t h a t t h e i n te r n a l d y n a m i c s a r e e x a c t l y t w o - d i m e n s i o n a l , a n d w e c h o o s e s p e c i f i c a ll yt o w o r k w i th t h e F i t z H u g h - N a g u m o e q u a ti o n s:

1 ( x - - a + b y ) -= f 2 ( x , y ) .= c ( x + y - x 3 ) - - f l ( x , y ) , Y = - - c ( 1 4 )

I t f o l l o w s t h a t t h e e i g h t - c e l l n e t w o r k i s 1 6 - d i m e n s i o n a l .I n o u r m o d e l s y s t e m w e a s s u m e t h a t a ll c o u p l i n g s a r e l i n e a r a n d d i a g o n a l , i n a se n s e t h at w e n o w m a k e p r e c i se .L e t ( x i , Y i ) b e t h e s t a t e v a r i a b l e s o f t h e i t h c e l l . W e a s s u m e t h a t t h e c o n t r a l a t e r a l c o u p l i n g s t r e n g t h c a n b e s e t a tv a l u e s d i f f e r e n t f r o m t h e i p s i l a t e r a l c o u p l i n g s t r e n g t h s . T h a t i s , w e a s s u m e t h a t

Jci = f l ( x i , Y i , ~ -) + o l ( x i - 2 - - x i ) "]- y ( x i + e - - x i ) ,Y i = f 2 ( x i , Y i , ) Q -}- f l ( Y i - 2 - - Y i ) + 6 ( Y i + c - - Y i ) , (15)

w h e r e i 1 . . . . . 8 , t h e i n d i c e s a r e t a k e n m o d u l o 8 , a n d+ 1 , i o d d ,

= - 1 , i e v e n .S e t t i n g a = 0 . 0 2 a n d b ---- 0 . 2 i n t h e F i t z H u g h - N a g u m o e q u a t i o n s w e f i n d t h a t s t a b le p e r i o d i c s o l u t io n s c o r r e -

s p o n d i n g t o e a c h o f t h e p r i m a r y g a i ts o c c u r a t th e p a r a m e t e r v a l u e s g i v e n i n T a b l e 5 . E a c h o f t h e s e s o l u ti o n s c a nb e f o u n d s t a r t i n g f r o m t h e i n i t i a l c o n d i t i o n ( X l, Y l ) = ( 0 . 0 6 , - 0 . 0 4 ) a n d ( x i , Y i ) = ( 0 , 0 ) f o r 2 < i < 8 . T hes i m u l a t i o n s v e r i f y i n g th e s e s t a t e m e n t s w e r e p e r f o r m e d u s i n g c l s to o l .

I t s h o u l d b e n o t e d t h a t e a c h o f t h e s e p e r i o d i c s o l u t i o n s i s f o u n d n e a r a s u p e r c r it i ca l H o p f b i f u r c a t io n o f t h ea p p r o p r i a t e t y p e , a s w e n o w d i s c u ss .


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M. Golubitsky et al./Physica D 115 (1998) 56-72Table 6Block d iagonalization of the linearization at the stand

71

Gait Representation Block diagonalizationPronk Uoo APace U01 A - 2CBound U2o A - 2BTrot U21 A - 2B - 2CJump U10 B A - B

( A - B - - 2 C - B )Walk Ull B A - B - 2C

6.1. Ho pf bi furcat ions

S u p p o s e t h a t z0 = ( x 0 , y 0 ) i s a n e q u i l i b r i u m o f ( 1 4 ) s o t h a t s e t t in g t h e i n t e r n a l v a r i a b l e s o f al l e i g h t c e l ls e q u a l t oz 0 is a n e q u i l i b r i u m f o r ( 1 5 ) . W e c a l l th i s e q u i l i b r i u m t h e stand a n d l o o k f o r H o p f b i f u r c a t i o n f r o m t h e s ta n d . T o f in dH o p f b i f u r c a t io n s , w e m u s t c o m p u t e t h e l i n e a r i z a ti o n o f ( 1 5 ) a t th e s t a n d , f i n d c o n d i t i o n s w h e n t h i s l i n e a ri z a ti o nh a s p u r e l y i m a g i n a r y e i g e n v a l u e s , a n d d e t e r m i n e w h e n t h e r e p r e s e n ta t i o n o f F o n t h e e ig e n s p a c e i s o f t h e ty p ec o r r e s p o n d i n g t o e a c h p r i m a r y g a i t . F r o m r e p r e s e n t a t i o n t h e o r y w e k n o w t h a t t h i s l i n e a r i z a t i o n i s b l o c k d i a g o n a lw i t h t h e b l o c k s c o r r e s p o n d i n g t o t h e s u b s p a c e s Uj k d e f i n e d i n ( 1 1 ) .

L e t

A = (d F) zo, ~ = c d ' ~ ' ~ "

T h e n t h e d i a g o n a l b l o c k s o f t h e l i n ea r i z a ti o n a r e g i v e n in T a b l e 6 . U s i n g t h is i n f o r m a t i o n o n e c a n c h e c k t h a t i n e a c hc a s e l i s te d i n T a b l e 5 , t h e p e r i o d i c s o l u t i o n f o u n d i s n e a r a p o i n t o f H o p f b i fu r c a t io n .

7 . C o n c l u s i o n s

W e h a v e a p p l i e d t h e t h e o r y o f s y m m e t r i c H o p f b i f u r c a ti o n t o i n f e r a p la u s i b l e c l a ss o f C P G n e t w o r k a r c h it e c tu r e sf r o m t h e o b s e r v e d p h a s e r e l a t io n s h i p s o f a n i m a l g a i t s. P r e v i o u s a t t e m p t s a l o n g s i m i l ar l i n es h a v e l e d t o n e t w o r k st h a t e i th e r g e n e r a t e t o o l i m i t e d a s e t o f g ai ts , o r s u f f e r f r o m t h e p r o b l e m o f u n w a n t e d c o n j u g a c i e s i n w h i c h g a i t st h a t a r e e x p e c t e d t o b e d i s t i n ct a r e r e l a t e d b y a s y m m e t r y a n d h e n c e h a v e i d e n t i c a l d y n a m i c a l c h a r a c t e ri s ti c s

W e h a v e s h o w n t h a t t h e se p r o b l e m s a r e i n e s c a p a b l e i f t h e g a i ts o f 2 n - l e g g e d a n i m a l s a r e m o d e l e d b y s y m m e t r i cC P G s c o m p o s e d o f 2 n i d e n t i c al c e l ls . H o w e v e r , t h e p r o p o s e d c l a s s o f n e t w o r k s w i t h 4 n c e l ls , w h i c h i s c h a r a c t e r iz e db y 77 2n x 772 s y m m e t r y , r e p r o d u c e s a l l o f t h e s t a n d a r d q u a d r u p e d a l g a i t s ( i n c l u d i n g t h e c a n t e r a n d t h e t r a n s v e r s e a n dr o t a r y g a l l o p s) , m a n y o f t h e s t an d a r d h e x a p o d a l g a i ts , a n d t h e t r a v e li n g - w a v e g a i t s / r h y t h m s o b s e r v e d i n c e n t ip e d e s ,m i l l i p e d e s a n d l a m p r e y . B e c a u s e t h e s y m m e t r y g r o u p i s a b e li a n , t h e r e a r e n o n o n t r i v i al c o n j u g a c i e s , h e n c e n ou n w a n t e d o n e s .

W h e t h e r t h e a r ch i t e c tu r e s o f a c tu a l a n i m a l l o c o m o t o r C P G s r e s e m b l e t h e p r o p o s e d n e t w o r k s r e q u i r e s f u r t h e ri n v e s t i g a t i o n . H o w e v e r , t h e s e n e t w o r k s h a v e p o t e n t i a l a p p l i c a t i o n s t o t h e c o n s t r u c t i o n o f l e g g e d r o b o t s . I n p a r ti c u l a r ,t h e y p e r m i t a m o d u l a r d e s i g n i n w h i c h t h e n u m b e r o f p a i r s o f l e g s o f a r o b o t c a n b e v a r i e d m e r e l y b y p l u g g i n gt o g e t h e r o r d e t a c h i n g t h e a p p r o p r i a te n u m b e r o f m o d u l e s . T h i s t y p e o f d e si g n s u g g e s t s n e w w a y s t o g e n e r a te a n dc o n t r o l t h e m o t i o n o f a d a p t iv e l e g g e d r o b o t s


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72 M. Golubitsky et aL /Physica D 115 (1998) 56-72Acknowledgements

T h i s re s e a rc h w a s s u p p o r t ed i n p a rt b y N S F G r a n t s D M S - 9 4 0 3 6 2 4 ( M G , P B ) a n d B E S - 9 6 3 4 0 2 4 ( JC ) , a n d t h eT e x a s A d v a n c e d R e s e a r c h P r o g r a m ( 0 0 3 6 5 2 0 3 7) ( M G , P B ) .

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A Modular Network for Legged Locomotion

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