A CPG Model for an Autonomous Decentralized Multilegged Robot

IFAC

Cop~Tight

l' IFAC Intelligent Autonomous Vehicles. Sapporo. Japan. 2001

[: 0 [> Publications w\\w .e lseyier.com 'locate ifac

A CPG MODEL FOR AN AUTONOMOUS DECENTRALIZED MULTILEGGED ROBOT Shinkichi Inagaki * Hideo Yuasa *' ** Tamio Arai ' * Graduate School of Engineering . Th p [inil'ersily of Tokyo.

Hongoll . Bunkyo-ku . Toky o. Japan ** Bio-Mimr:tic Control R eseach CeT/ter. RIA'EN. ATll1gahora.

Shi1l!0-shidami. Moriyama-ku . Nagoya. Japan

Abstract: A central pattern generator (CPG ) model is proposed for a walk patt em generation mechanism of an autonomous decentralized multilegged robot . The topological structure of the CPG is represented as a graph. and on which two timeevolution s~'s t ems: a wa\'e equation (Hamilton s~'st e m ) and a gradient s~'st e lll are introduced. The CPG monel can generate osc illation patterns depended onl~' on the network topology and bifurcate different oscillation pa ttem s according to the network energ~·. This means that the robot can generate gait patterns onl~' b~' connecting legs and transit between gait patt erns according to a parameter sllch as a dri\'ing intention of locomotion. Copyright © 200] ]FAC K e~'word s : Decentra lized s~·s t ems. Oscillators. :\etwork topologies. Patt crn generation. Transition modes. \Ya\,es. \\'aw equ ations

1. I:\TRODUCTIO:\

a nd utilized to coordinate leg movcments (Fig. 1) (Taga. 199.5: Schoner et aJ.. 1990: Yuasa and Ito. 1990: Ba\' and Hemami. 1987: Coli ins and Richmond. 199-1: Cana\'ier et al .. 1997) .

Legged animals can generate suitable walk patterns (gait s) for the wa lk speed. This gait generating s\'stem is basica ll~' achie\'ed in a central pattern generator (C PG ). \\'hich is a group of neurons loca ted in a central nen 'ous s ~·s t e m ( P e ar so n . 1(76 ).

Gait transitions are produced by selec tiwl~' changing either the relatiw strength (Schoner et aJ.. 1990: Yuasa and Ito. 1990 ) or the polarit\'(Ba~' and Hema mi. 1987 ) of the coupling between oscillators. The CPG models need to be re\\'ired esse nti a ll~' in order to make a different walk pat-

An a utonomous decentralized nlllltilegged robot has each leg as an a utonomous partial s ~' s t e m (s\lbs~'stem) \\'ith an oscillator(Odashima et aJ.. 1(99) . This robot has the function of a CPG as the \\'hole s~'stem b~' coupling \\'ith neighbor oscillators. i.e. coupling lo ca ll~·. Each oscillat or is assumed not to be able to get information about the total number of oscillators or its position in the net\\·ork. It is con\'ersel~' expected that s uch a robot s~'stem ha\'e properties such as extension of legs. eas.\· maintena nce. fa ilure- tolerance. and en\'ironmental adaptabilit~·.

Ll~Rl

L2

R2

L3

R3

I't!J)l Suboystem

The CPG is 11l0deled as a s~'st em of coupled nonlinear oscillators because of the periodical output.

Fig. 1. The schema of a CPG

255

@l> motor neuron

tern \"ith these modeling methods( Collins and Richmond, 1994). This means that each phase relation of common gait patterns needs to be embedded into the connections between oscillators according to the number of oscillators. This is contrary to the desired property 1. On the other hand. Collins and Richard(Collins and Richmond . 1994) haw proposed a hard-wired CPG model. The CPG model is composed of oscillators with fixed internal coupling. However, the CPG model can generate various phase-locked oscillating patterns equivalent to common gait patterns. In this modeling scheme, gait transitions can be produced by changing the intemal parameters of an oscillator or continuous, periodic inputs( Coli ins and Richmond , 1994: Canavier et al.. 1997). However. it is required to allalyze on how to change the parameters and inputs in various cases.

tion mode emerges alternatively out of multiple ones and the oscillating mode transits against the change of the network energ~·. In Sec.5. computer simulations of a hexapodal CPG model is demonstrated.

2. THE HA:\IILTO:\, SYSTEi\1 O:\' A GRAPH

2.1 Definition of graph and function spaces This section is started with some definitions about complex function on a graph. Let a set of vertices be V. a set of edges be E. then a graph G is defined as a set of these two terms G = (V , E) (Fig. 2). It is assumed that G is a finite and an oriented graph. That is, the number of edges is finite . Each edge connects from the initial vertex to the terminal one. This is because codifferentiation or gradient defined on the graph(Urakawa, 1996: Yuasa and Ito. 1999). Note that this direction is not related with the information flow , that is, the interaction of each node is bi-direction.

In this research. a CPG model is expressed as a graph by an oscillator corresponded to a vertex and an interaction between oscillators corresponded to an edge (Fig. 2). Thereby, the gait patterns result in the eigenvalue problem of graph. i\Ioreover, each gait pattern has an eigenvalue of the graph as an intrinsic energy value. Gait patterns inhere in the way to connect oscillators. and gait transitions are achieved only by varying the network energy.

Let C(V) be the set of all complex functions on V. Laplacian operator L':.A on a finite graph G is defined as follows:

As a result , the multilegged robots with the arbitrary number of legs have following properties: v""-'u

(1) Gait patterns can be generated automatically according to the number of legs without embedding gait patterns. (2) Transitions of gait patterns can be generated only by changing a parameter, i.e. the network energy.

for '11' E C(V) (Urakawa, 1996). Where deg(u) denotes the degree of vertex u, that is, the number of connected edges of vertex u, and v "-' u denotes that the vertices l ' and u are connected(Fig. 3). Note that L':.A is calculated as follows: (3)

The organization of this paper is as follows . In Sec.2, we define a graph and a functional space, and then introduce wave equation that forms Hamilton system on the graph and obtain inherently multiple natural oscillation modes. :\loreover, a method of derivation Hamiltonian and natural oscillation modes is shown. In Sec.3. the derivation of natural oscillation modes in the graph with six vertices is demonstrated as a simple example. An application of the graph as a CPG model and a relation between the natural oscillation modes and gait patterns are discussed . In Sec.4. Hamilton system is synthesized with the gradient s~'stem so that one natural oscilla-

where Aa is an incidence matrix of G(Ozawa, 1980) . This incidence matrix Aa is determined by connections of verteces.

2.2 A wave equation as Hamilton system It is introduced that Schrodinger's ,,,ave equation on graph G , which is one of Hamilton system. Lett·(t) = (t:,(u. t))uEV E C(V x R) be a wave function on graph G . Then. SchrOdinger's waw equation on graph G is defined as follows:

oo( t) ih----;)t = L':.Al'(t)

+ FL, (t).

(4)

o:

subeyBtem inter&etion

Fig. 2. Graph expression of a system

Fig. 3. Vertex

256

II

and its neighbor verteces

l'

where h plays the role of Planck's constant \vhich relates Hamiltonian and angular velocit~·. F is a potential operator which prescribes boundary condition of this system. The first term of the right hand side in eq.(4) is positive because of the definition of Laplacian operator on a graph in eq.(2) which is against to that of Laplacian operator on a continuous s~·stem.

t,,(t) is expressed b~' the linear sum of the natural frequency modes: 1\'-1

c(t)

=

L km!;'m(t)

(11)

m=O

where km E 'J are mode coefficients.

Here. it is assumed that the potential operator l,r is limited as F = O. Then. each d~' namics of eq.( 4) at each vertex is expressed as follows :

3. Ai\"

EXA~IPLE

: A GRAPH WITH SIX VETICES

(5)

3.1 The natural oscillation modes that appears in a graph of six vertices

This equation shows that each dynamics at each vertex is determined by values of neighbor vertices and itself (Fig. 3).

In this section, we deal with a graph composed of six vertices as shown in Fig. 4. In this case, the incidence matrix Aa is as follows:

ih

al'(U t) Of' = -

Ld v . f) + deg(u)t',(u. f) v""-'u

Hamiltonian operator H is defined as H Then , eq.(4) becomes ih a'lj,(t)

at

=

HIi,(t) .

-1 1

== llA . Aa =

(6)

This means that Hamiltonian H of this Hamiltonian operator H is equal to the energy of this network. This Hamiltonian H can be decomposed as local value H(u) at each vertex u.

L

uEV H(u) =Ii'*(u. f) (Hv,(t)) (u)

(8)

2.3 Natural frequency mode of wave Natural frequenc~' modes of the wave equation (4) are solutions of the following equation. (9)

That is. the natural frequency modes t"m (t) (m = O. 1. 2.. . .. N - 1) are provided as characteristic functions of Hamiltonian operator H . Then Hamiltonian Hm of Hamiltonian operator Hare eigenvalues of them . To use the separation of variables. these natural frequency modes of eq.(9) are expressed b~' the function of verteces ym = (ym(U))uEI' where lI:'::mll = 1. and the function of time Om(t) = yme-i~·",t

(12)

The natural oscillation modes obtained from Fig. 4 are matched with hexapodal gaits as shown in Table l. where each gait pattern has phase differences between the legs as shown in Fig. 6. Here. we assume that the natural oscillation modes. with a same eigenvalue. form one gait b~' exchanging through one c~'cle of the oscillation.

e-i~·",t.

= ymOm(t) =

0 0 0 0 0 0 0 1 o -1 0 o -1 0 1 0 0 1 -1

Here. we consider that a robot with N (even) legs has the CPG model composed of 2N oscillators as shown in Fig. 5 (Odashima et aL 1999: Golubitsky et al.. 1998). We arrange two networks symmetrically, and assume that oscillators 0.2. · ·· . N -2 and 0',2'.··· . (N - 2)' control timings of the legs Ll.L2,.·· , L~ and R1 ,R2,··· .R~ . Here. we assume that the legs of each segment, i.e. L3 and R3. are half a period out of phases seen in common gaits of insects(Odashima et al.. 1999) . Thell. hexapodal gait patterns are generated as shown in Fig. 4. The reason for employing the structure are CD the locality of the connections of each oscillator. and @ the generation of a traveling wave in both sides. For example of @. a traveling wave through the network can generate a 'metachronal tripod' in hexapodal gaits and a 'walk' in quadrupedal gaits.

H (u) is also determined by values of neighbor vertices and itself.

C'm(t)

0 0 0

0 0 1

3.2 The natural oscillation modes that appears in a graph of six vertices

(7)

H(u)

o -1 0 0 0

0

Eigenfunctions :'::m and Hamiltonians Hm are obtained from Eq.(9) as shown in Table l.

H =l/,*(t) ·H1b(t)

=

1

o -1

(10)

\Vhen Wm is a natural angular frequency, it is related with Hamiltonian Hm as Hm = hwm· In this Hamilton system. an arbitrary waYe function

Interestingl~'.

the larger becomes Hamiltonian Hm . the more rapid are these gaits in real insects.

257

Table 1. Total Hamiltonian and Eigen-function of Hamilton matrix

_ 'Pm -

['_(0 1 ]

'Pm( l ) 'Pm (2) 'Pm (3) 'Pm(4) 'Pm(5)

gaits

(a)stand )4-~-{

(q(4, t) ,p(4,t))

~ [j1

4

1

1

[=i]

"2

3

2 4

1 4

0 0

m Hm

1

"2

4 12

12

[=!]

1

1

"2

5 24

"2

[=i1 [=!l (c) rolling tripod

(b) metachron a l tripod

v'6 """"6

[:)]

(d)tripod

A scenario that rea lizes (a).(b) was considered as follows:

(q(J,t),p( J, t))

(a) A varia ble of each vertex converges to a unit circular orbit in a complex pla ne. Then s uch a natura l oscilla tion mode appears t hat is the closest to a t arget value of Hamiltonian. (b) When the t arget value of Ha miltonian increases (or decreases) co ntinu o u s l ~ ' , the variable of each vertex follows it by enlarging (or reducing) the orbita l radius at first . If the a mouut of change becomes larger , the value of each vertex changes (bifurcates) t he nat ural oscilla tion mode. and cont inues the follow-up t.o the t arget value( Fig.7) .

5 (q(5,t),p(5,t))

Fig. 4. A graph of 6 nodes

Let :.r(t) = (x(1£. t))"EV be a varia ble on a vertex. If a potential fun ction is represented as P = P (:r (t )), a dy namics of t.he gr adient s ~ 'ste m ( Yu asa and Ito. 1999) is expressed as a:.r( t ) at

-i 1

o

L3 - R3 0 3/ 6 L3 - R3 (a)stand

P (.T( t ))

(b) meta.chronal tripod

3/6 L3 - R3 (c)rolling tripod

=

L

P(x( 1£ , t ), J"('u . t )lv "'"' 1/)

(14)

uEI-'

5/6~J _~1 1/6~1 -~J 1/2~1

/6 /6 0 L2 - R2 0 4/6 L2 - R2 /6 5/6 L2 - R2 /6

(13)

If the pot.ential function P is a s um of local pot ential fun ctions defined on each vertex

Fig. 5. The str uct ure of a N legged robot with 2N oscill ators in one side: a region in a broken oval is treated as one mod ule

O~l o

<5P (:r( t )) ox( t )

Eq.( 13 ) is decomposed into each vertex:

- i 10 0 L2 - R2 1/2

a ;r(t) 11 at ( )

1/2 L3 - R3 (d) tripod

=_

+L

Fig. 6. Phase relations of hexapod al gaits

( OP(:r( lI. t ), ;1'(1', 0 11' '" 11 ) (h:(t ) <5 P (.T(l', t )',:r(uo ,t )luo

l'~"

-1 . SYl\THSIS OF HA~IILTO:\ SYSTE.\I A:\D A GRADIEl\T SYSTE1-.1

"'"' V) )

(1/)( 15)

OJ'( t )

Then it is needed to be assumed tha t each ver tex can obtain the information of t he vertices to two edges further. However. it depends on how to choose the potential fun ction . The equa tion differentiated of Eq.( 23 ) ma ~' be decided only b~' the informa t.ion of itself a nd neighbor vertices. Such a pot ential function is chosen in Sec.4. 2.

4.1 A design of a gradient system Ha milton s ~'ste m merely produ ces a linear sum of the oscillating modes as demonstrated in Sec. 2. A dynamics as follows: (a) . (b) is realized via synthesis of Hamilton syst em and a gradient s~·s t el11 .

4. 2 Th e synthesized system

(a) One (or tv,:o) specific natural oscillation mode(or modes) appears a ltern a tivel~' o ut of mult iple natural oscillat ion modes. (b) Changes between gait pa tterns are performed s uit abl~' by a parameter change, such as walk speed.

A dy namics of a as a U' (t)

s~' nthes ized

- .- = Ut

258

i

- -H~'( t )

h

syst.em is expressed

JP

- -, -

o ~, (t)

(16)

~(O,t)

-1

1

I

~"'(O, t)

Hl_-l

-1

(

model·2

Fig. 7. Transition between nat ural oscillation modes with a target value HI Ta ble 2. the para meters ({ 5

b

r

h

Hamiltonian HI was

d

HI

0.0..1

Ta ble 3. the illit ial sta tes 0.95

+ iO.S5

li'( l. 0)

0.02 - i O.!:J5

v(2,0)

0.9..1 - iO.93

'1.',( 3.0)

- 0.3 1 + iO.58

d ..t.O)

0.72 - iO.4 1

1.-'(5.0)

0.88 - iO. 05

'1 .'(0',0)

O, Ti - iO.07

'IJ!( l ',O)

0,57 - iO,97

'u !(2'. 0 )

0, 93 - iO,2 1

~ ,'(3', 0 )

- 0.3 7 - iO.5 4

'1/'( ..1 '. 0 )

0, 3G - iO,5(i

~ ' (.5', 0)

- O, Gl

In Fig. 8. the rea l part of the vari a ble Oll vertex No.O is plotted vs . t ime (t = 0 '" 400 ). The ('~; cle of the oscillating pat tern changed dr alll ati ca ll ~' at near t = 100 and t = 350.

+ iO.O..!

T he excimnge-patteru of the modes 1 · 2 occupied until t = 100. T hen the pa ttern tra nsited to the exchange-pat tern of t he modes 3 . 4 at t = 100. },Ioreover. the exchange-pa t teru of the modes transited from 3·4 t.o 1 · 2 again a t around t = 350 while H I is red ucing.

from equ a tion Eq.(6) a nd (13) . In this research . a pot,ent ial fun ction s uch as (17 ) is chosen. \-vhere

L

P".(v) .

= 1 .2 or 3

k

Fig. 9 is a plot. of )Rt'tu. t)( 11 = O. 2. 4 a nd 0' . 2'.4') at t = 345 '" 365. T he number above each curve expresses the number of the correspondillg oscillator. T he former and later oscill ation patterus correspond rolling tripod and metachronal tripod res p ec ti ve l~' from Ta ble l.

(18)

HE V

and ,

?

b

Po(v)

= (11 d u. t) 11 - 1 + bt + 11 I."(v. ' t ) 11"- (19)

PI (1/ )

= (H (lI)

P'2(u)

= P'2( lI ') =11

(I{ n. t ) + t"(7t'. t )

Fig. 10 shows the changes of Hamiltoni all H (the solid lille) and the target value of loca l Hamil tonian H I (t he broken line). where H is ca lcul ated bv Eq.( 7). It is clear tha t H changed d rama tically between the excha nge-patterns of t he modes 1 ·2 and 3· 4.

(20 )

- HI )'2

11'2

(22)

as shown in Fig. 10 (the d ashed line) . The result of this simulation is shown in Fig. 8", Fig. 10.

'1."(0.0)

H· =

= 6 (1 - cos (2r.t / 400))

(2 1)

from (a) , (b) of SecA. l. where a . b . c a nd cl are rea l constant s, and H I is a t arget va llle of local Ha milt oniall H (ll ). Po works so t hat the variable of each vert ex converges to a circular orbit with ra diu s b in a com plex plane. P I works so that the loca l Ha miltonia n H (ll ) of each vertex follows t he t arget value HI . P'2 works so th at waves in left a nd right networks become a ntiphase.

It turns out tha t the proposed C P G model can achieve t ransitions between oscillat ing pattem s against the cha nge of HI ' Furthermore. if t he t arget va lue of local Hamilt onian H I is ma tched wit h walk speed. the gait patterns would be cha ngeable against the walk speed . Ext ractions of specific modes and transitions a re observed in simulations of 4 legs and 8 legs \vit h the sa me paramet ers of Table 2 """hen t he para meters and t he initial condition are different. however , modes \vith the different angular frequencies m a~' be intermingled or a transient stat e may become longer. This problem is proba bly caused by the indefinit e of t he mechanism to extract a particular mode. As the fu t ure s ubject, elucid ation the problem and est ablishment a methodology for designing a suita ble potent ia l fUll ction are needed.

5. G AITS GE:,\ERATIOf\ AND TRA:\,SITIOf\ Il'\ CO},IPUTER SH-IULATIOl\S \\'e simula ted generation a nd transition of hexapoda l gait s with t he connections as shown in Fig. 5. The para meters in Eq. (16 ) are shown in Table 2. and t he initial values are shown in Table 3 (a real part a nd an imaginary part of a initial varia ble were generated respectively by an uniform ra ndom number of [- 1, 1]) . The target value of local

259

and \yalk speed was implied. Howewr. there is no theoret ica l corroboration of the correspondence. T he rela tion of Ha miltonian a nd ,,-a lk speed " 'ill be dpclared bY matching nat ural osc illat ion modes and snllmetries. :\loreon'r. fo llo,,-s are tlw future research subjects:

~v(0 .

Fig. 8. The waye forms of

• a mat hemat ical background a bou t s~' nt hesis of Halllilton s~' s t e m and a gradient system . • const ru ction of a proper potential function to gellerate desired natu ra l oscilla tion modes. • other applications of o ur method .

t)

I~:~-.-~ ~: ~ -~-:-:. - - -~ ~: r: I

1.5":=~/~/:lL'=:;:::======-----, References

F ig.

9. The wave 0,2 . 4.0'.2',4')

~ ~'(u .

forms of

.J . S. B a ~- alld H. Hemami. :\lodelillg of a neural patt em generator with coupled nonlinear oscill at ors. IEE E Transactions 071 Biom edical E ngin eering. B\lE-34--1:297- 306. 1987. C. C. Ca ww ier. R. J. Butera . R. O . D ror. D . A. Baxter. J. \Y . Clark. and J. H. B~Tn e. Phase response characteristics of model lleurons det ermine which pattem s are expressed ill a ring circuit model of ga it generation . Biological Cybernetics. 77: 367- 380. 1997. J . J . Collins and S. A. Richmond . Hard - wired centra l pattern generators for qu adrupeda l locomotion. B iological Cy be1il.etics. 71: 37.5- 385. 199-1 . \1. Go lu b its k~·. 1. St ewart. P . L. Buono. and J. J. Collins. A mod ular network for legged locomotion . P h ysicaD. 11 5:56- 72. 1998. T . Od ashima . H. Yuasa. Z. " -. Luo. a nd ~1. Ito. Emergent generation of ga it pattem for a m~T­ iapod ro bot sYst em based on e n e r g~' consumption . J ournal of th e R obotics Soc iety of J apan (in J apanese ). 17-8: 1149 11 57. 1999. T. O Za\ya. Electric Circuit I (in J apanes e). Shokodo. TokYo. 1980. K . Pearsoll _ The cont rol of walking. S cientific A.m erican. 235-6: 72- 86. 1976. G . Schoner. \Y . Y ..Jiang. a nd J. A. S. Kelso. A s~- n e r ge ti c th eor~' of quad ra ped gaits and gait tra nsitions. J o urnal of tli eore tica l biology. 1-12: 359- 391. 1990. G. Taga . A model of the neuro- musculo-skeletal s~'s t em fo r human locomotion i. emergence of basic ga it. Biological Cy bern etics. 73:97- Ill. 1995. H. Uraka\ya. Laplacian an d X ehrorks (in J apanese ). Shoka bo. Tok~-o . 1996. H . Yu asa and :\1. Ito. Coordination of m a n~- oscillators a nd generation of loco mo t or ~ ' patt erns. B io logica l Cy bern etics. 63: 177- 18-1. 1990. H. Yuasa and \1. Ito. Aut onomous decentra lized s ~-st ems a nd reaction-diffusion equation on a graph . T ran s action s of th e Society of In stru -

t)(u

+---; --- ~ ---; ----i- ---; ---i~~ :~~~:~:: : ::J~:J~:~L:j::::'::::

30.-~--~~--~--~~--,--.

a25 - - - - i---

ex:

5

o

' ,,.

",

'

I

I

.....

'

I

---,--r :----'----:----,----~ --,----

50

,

100

150

200 t

250

300

.....

~

350

400

Fig. 10. Total Hamiltonian H and the target loca l Ha miltonian HI 6. DISCUSSIO:\ In this pa per. a method t o constru ct a m athema tica l C P G model was proposed . The d~' n a mics of each osc illator is determined only b~' information of it self and neighbor oscillators. :\('yertheless. the s ~'s t em has a nat ura l oscillation mode as a tot a lorder. The order depend s on a sca le of the s\·stem . i. e. t he number of oscillators. As a result . we can prod uce multilegged robots with the a r b itr a r~' number of legs. The ro bots haw following properties: (1 ) Ga it pa tterns can be genera ted a uto matica lly according to the number of legs ,,-ithout embedding ga it pa tt erns. (2) Transitions of gait patt erns ca n be generated onlY by cha nging a parameter. i. e. the e n er g~' of the osc illator net,,-ork. In this paper. a "'a ~' of each leg mowment aga inst the yaria bles of the oscillator was not trea ted. TIH' correspondence is required in order to carry the C PG model in a n act ua l robot . :\lorem-er. a correspondence \yit h the actua l ph~ -s i ca l s~-s t e m of a t arget yalue of a local Hamilt onian HI is also needed. In Sec.3. correspondence of Hamiltonia n

m e nt a nd Con t rol En gineers (in J apa nese) . 3.5:

1-1-17- 1453 .1 999.

260