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C o n t r o lo f a Q u a d r u p e dTro t

J S

Glower nd U. Ozguner

..

..

..

D e p a r t m e n t o f E l e c t r i c a l E n g i n e e r i n g

The Oh ioS t a t eU n i v e r s i t y

Columbus, Ohio 43210

ABSTRACT

A s i m p l ec o n t r o l a w sp r o p o s e d om a i n t a i n

t h e m ov em en t o f a t r o t t i n g q ua dr up ed .he

a s s u m p t i o ne h i n dh i so n t r o la wshat one

s h o u l d b e a b l eoe l y u po n t h eu a d r u p e d ' s

m om entum f o r t a b i l i t y . The c o n t r o la wh o u l d

o n l ye r v eoe d i r e c tra i n t a i nh e

quad rupe d 's momentum. The r e s u l ts a c o n t r o l

l a wh a tw o r k swe l l so l on g as the uad ruped i s

movi ng

INTRODUCTION

Thereas ee n a l o t f e c e n t n t e r e s t n

wa lk ingm a c h i n e s a nd r e s u l t s h av ee e n r e p o r t e d

on b o t hs i x [1,2] a n d o u r [33 l e g g e d o c o m o t i o n .

I n

t h i s p ap er , a c o n t r o la wh a t t a b i l i z e s a

q u a d r u p e d t r o ti t h o u te l y i n g upon s t a t i c

s t a b i l i t ys p resen ted . I n o r d e rot a b i l i z e

t h e q u ad ru p ed , f i v ee p a r a t en p u t sr eo n -

s i d e r e dh a tf f e c th eu a d r u p e d ' su t p u t s :

v e l o c i t y , e i g h t , o l l , e l e v a t i o n , a n dzimuth.

T h r e eu t p u t sf o r w a r de a n ,a t e r a lean , and

1a t e r a l e l o c i t y ) r e o t i e c t l y o n t r o l e d

s i n c e h e ya r e b ou nd ed o r i m p l e r a n s f o r m a t i o n s

o fo t h e ro u t p u t s .

T h e a s s u m p t i o ne h i n dh i so n t r o la ws

t h a t o ne s h o u l d b e a b l e o e l y u po n t h e q ua d-

r u p e d ' s m omentum f o r t a b i l i t y . T he c o n t r o la w

s h o u l d n l ye r v eoe d i r e c t rm a i n t a i nh e

quad rupe d 's momentum. The r e s u l ts a c o n t r o l

l a wh a tw o r k sw e l l

so

l o n g as t h e u a d r u pe d i s

moving.

QUADRUPED MODEL

Them od el f o r a q u ad ru p ed s e d i n h i s p a p e r

i s a s i n g l e i g i d mass i n 3 -s pa ce f o r a b ody w i t h

f o u rx t e n s i b l ee g st t a c h e doa c ho r n e r .

Each l e g sa b l e og e n e r a t e w oh i p o r q u e s

and

o n eo r c el o n gh ee n g t hfh eeg . The

r e s u l t a n t o r c e s a nd t o r q ue s

on

t h e b o d yproduced

b y e g

j

i n body oo rd ina tes re as fo l l o ws [43:

Top View

Side

View

t Y l

Front iew

F i g u r e 1. QuadrupedModel

1496

CH2282-2/86/0000/1496 01.00986BEE

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I f one ssumes ma ss les s e g s , h e n h e e s u l t a n t

a c c e l e r a t i o n so f h e body i n b o dyc oo r d i na t esc an

b e o u n d r o m h e o l l o w in ge q u a t i o n s :

1-

T =

where ~ j ds h e o t a t i o n a l n e r t i a a b o u t h e t h

a x i s , g s h ea c c e l e r a t i o n d ue t og r a v i t y , a n d

T 1 s a 3 x 3 r a n s f o rm a t i o n r o me a r t hc o o r d i n a t e s

t o b od ycoord ina tes .

I nb t a i n i n gq u a t i o n s 3 and

4,

mass less

l egsw e r e as sum ed so t h a t h e e r i v a t i v e

a~

i s

s i m p l y

Q W I f

onea s l e g sw i th mass, th en h e

r o t a t i o n a ln e r t i a i s no longeron sta nt . As a

r e s u l t , a mo re c o m p l i c a t e d o r mo f h ee q u a t i o n s

3 and

4

a r e equ i r ed . As a f i r s to r d e ra p p r o x i -

mat ion, owever, one can assume t h a t h e change

i nh e r o d u c t fo t a t i o n a ln e r t i a and r o t a -

t i o n a le l o c i t ysm a l l .i n c e a t r o t t i n g

quad r uped i se i n go n s i d e r e d ,h eo t a t i o n a l

v e l o c i t yh o u l d be n e g l i g i b l e

i f

t h er o ts

s tab le . Thus ,ne may assume t h a td d i n ge g s

w i t h mass will n o tf f e c tq u a t i o n s 3 and 4

ex ces s iv e ly . However , th e max imum speed th a t a

quad r upedanr av e l wi b e a f f e c t e d .na c t ,

o n e h o u l di n dh a t a s t a b l e r o t x i s t s n l y

f o r i m i t e d speeds.

GAIT COORDINATION

T he u rp os e o f h i s w or k i s o i n d a c on -

t r o l a w o r h e e g o r c e s and t o rq u ess u c h h a t

a quadr uped i s a b l e o r o t a t a s peed f o rw h i c h

c o n c e p t so f t a t i c t a b i l i t y may n o t b e u s e f u l .

T he i n p u t s o h e s y s t e m r e h ee g o r c es and

t o r ques . As a r e s u l t , one m us t av eegs nder -

ne ath he body some o f he im e op r o v i d e h e s e

c o n t r o lo r qu es and f o r c es . The pu r po s e f a i t

c o o r d i n a t i o n s o see t h a t u p p o r t i n g e g s r e

a v a i l a b l e o s u p p o r t h e b od y.

T h e g a i to n s i d e r e dnh i s a p e rsh e

s i n g u l a r r o t 151 I n a s i n g u l a r r o t , one p a i r

o fd i a g o n a l l yo p p o s i t e e g sa r ea l w a y ss u p p o r t i n g

t h e body. When one p a i r f e g s s i f t e d , h e

o t h e r a i rse t down. The g ai to o r d i n a t i o n

a l g o r i t h m ,h e r e f o r e ,u s te t e r m i n e when t h e

s u p p o r t i n g e g sa r e o be s w i t chedandwhere he

new pai

r

o fs u p p o r t i n g e g sa re o be s et down.

S in c e a s i n g u l a r r o t s a s y m m et r ic a i t, a i t

c o o r d i n a t i o ns i m p l e :h e u p p o r t i n ge g s r e

s w i t c hed when t heu p p o r t i n gegs move h a l f a

s t r o k eb e h i n d h e i r e s p e c t i v ehi ps . The new se t

o fu p p o r t i n ge g sr ee t down h a l f a s t r o k e

ahead o f he i

r

r e s p e c t i v e h i p s .

It

h as t o b e m e n t i o n e dh a t ,w i t hm a s s l e s s

l e g s ,s u p p o r t i n g e g sca n be se t down anywhere a t

anyime . When t h eegs havemass,owever , the

l e g d y n a m i cs r ec l u d em m e d i a t ee g o s i t i o n i n g .

When a l e g s n t s w i n g p ha se , it shou ld be

d r i v e n o h e o r r e c t t a r t i n gp o s i t i o n np r e -

p a r a t i o no r e c o m i n g a s u p p o r t i n ge g .nh e

s i m u l a t i o n , h i s sa c c o m p l i s h e db yd e f i n i n g h e

f o r e - a f t o r q u e o f e g j t o be:

where A i s h e e g a n g l e h a t p l a c e s o o t j h a l f

a s t r o k e a he ad o f t s h i p p o s i t i o n .

ATTITUDE CONTROL

G i v e n h a t h e r ea r e u p p o r t i n g e e tu n d e r -

n e a t h h e b od y, t h e u p p o r t i n g e g ' sh i p o r q u e s

an d f o r c e s h a t t a b i l i z e h eq u a d r u p e d ' sm o t i o n

h av e t o b e d e f in e d .F o r h e a k eo f onv en i enc e ,

t h e i po r q u e s an d f o r c e s

w

11erouped as

f o l l ows:

s

= s 1 s 2 + s3 + s4

M =

M 1

+ M2

+

M3 + M4

F = F 1 + F2

+

F3 + F4

As =

+

s2

- s3 - s4

AF

=

F 1

+

F2

-

F3

-

F4

Assuming

= 1 s i n ( a )

Y

=

1 s i n ( @ )

c o s ( p )

= 1 6 )

equa t i ons

1

and

2

c a n b e w r i t t e n a s :

1497

7/25/2019 Control of a Quadruped Trot

3/6

Themodel g iven i n Eq. (7 ) imp1 esw e l v e

degrees fr e ed o m t o b e c o n t r o l l e d .This number

canbe educed t o n i n e

i f

t h e qu ad ru pe d i s o r c e d

t o f a c e h e i r e c t io n fm o t i o n . Assume t h a t X,

t h e o r w a r d e a no f h eq u a d r u p e d , s i m i t e d o

f

1 / 2t r ok e . If h e q u a d r u p e de a n so oa r

f o r w a r d , hen ae w p a i r

pf

s u p p o r t i n g e g s

wi

be et down. Fur the rmor e

Y,

( t h e a t e r a lv e l o c i t y

o fh eu a r u p e d )so r c e doe r o b y

0.

I f

t h e r es some l a t e r a le l o c i t y ,h e nh e

quadruped will t u r n o c a nc e l u th i sa t e r a l

v e l o c i t y .i k e w i s e ,

Y

( t h ea t e r a le a n fh e

quadruped)

will

be smal lue t o

Y

be i ngm a l l .

s i n c eh eu a d r u p e ds o v i n g ,h eu p p o r t i n g

f e e t

will

behanged f a i r l yr e q u e n t l y . As a

r e s u l t , h e m a l l r r o r n

Y will

no t av eime

t o n t e g r a t e o a s ub s ta n t i a l a l ueb e f o r e aew

p a i ro f e g sa r ep l a c e d ahead o f h e i r e s p e c t i v e

h i p s .

S i nc e n l y i ne eg r ees fr eedom eedo

b e o n t r o l l e d , q u a t i o n7 ) may be r e w r i t t e n n

s t a t e s pa ceo r m as f o l l ow s :

+

1498

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - -

* F

AS

0 0 0

F

n n

- x / 1

/cos

B

)

0

- ~ C O S ( U ) C O S ( B )

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Y I C O S a / 1 0

0

_ _

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - - - -

0

0 0

0

- 5 C O S ( p ) / i5 Y C O S ( a ) / l

-

8 )

The fo l low inga r a m e t e r sr eo n s i d e r e dor a

quadruped:

A

= 5 f e e t , B = 2 f e e t , I:

=

1 f o o t ,

oxx =

166 .66 s l ug ee t , @yy

=

866 .66 s l u g ee t ,

aZz =

966.66 s l ug ee t ,

m = 1

s l ugs ,

g = 32 . 2 f t / s ? ,

1 =

6 f e e t

I J s i n gh e s e a r a m e t e r s ,h ei n e a r i z e do r m f

equa t i on

( 8 )

becomes:

x

= 5.360

x +

5.3608

-

0.001667M 0.004629AF

8 =

.6192

X t

3.7158

-

0.001154M

+

0.005769~F

..

=

3.220

Y +

19.32

@ +

0.012000

S

Z

= 32.2 - Xoa

+

0.0092F

. .

4 =

3.181 4

+

0.000862

A S 9 )

CONTROLAW

The purposefh eon t r o l 1 aw i so

s t a b i l i z e a q u ad ru pe d t r o t . o rh e i m u l a t i o n

r uns ,he uadrupe d was moved a t a speed o f

20

f e e tp e r s ec on d w i t h a s t r o k e e n g t ho f 10 f e e t .

T h i s m p l i e s h a t a ew p a i r o f e g s

w i

11 support

t h e body ev ery 0.5 seconds. As a r e su l t ,h e

quadruped 's o rward ean,

X

v a r i e s a s a saw too th

s i g n a l i t h an a m p l i t u d e o f 5 f e e t and a f r e -

quency o f 2Hz.

I f

t heq u a dr u p ed i s om a i n t a i n a

s t a b l er o t ,h e nh eo i s en p u t

X

m us to t

c a u s ex c e s s i v eit ch in g. One way o fs s u r i n g

t h i s s o e q u i r e h a t h eg a i n r o m

X

t o

e

i n

e q u a t i o n

9 )

be smal l .ssum ing

10

degree

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o s c i l l a t i o n s o be t h ea r g e s t c c e p t a b l e ,h e n

t h eg a i n r o m X t o e s hou l d ee s shan -25dB:

The i npu t s

M y S,

F,

A S a n dA Fa r es p e c i f i e d

a s o l l o w s :

. .

M =

-924(X

-

X

-

26240

e

10496

e

S

=

-12000

Q

-

2400

Q

F = .-2OOO(Z - Z o )

-

4000

?

AS = -200000 I, 4000

. .

F

= -26230(X

-

X

-

131200 e

-

52480

e

( 1 0 )

theneach ubsys tem i n e q u a t io n 9 ) i s s t a b l e a n d

t h e a i nr o m X t o

e

i s -28dB .hus, t h i s c on-

t r o la w a pp ea rs t o b e a c a n d i d a t eo r

s t a b i 1i i n g a q u a d r u p e d t r o t .

A

second way o f hec k ingw h e t h e rh e o n -

t r o la w will s t a b i l i z e

a

q ua dru pe d t r o t s o

use a wors t ase pproac h ia yapuno v 's method.

The f o l l ow i ng pp r ox i m a t i onswere made i n equa-

t i o n ( 8 ) :

0.76