control of a quadruped trot

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  • 7/25/2019 Control of a Quadruped Trot

    1/6

    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

    Authorized licensed use limited to: Khajeh Nasir Toosi University of Technology. Downloaded on December 21, 2009 at 05:11 from IEEE Xplore. Restrictions apply.

  • 7/25/2019 Control of a Quadruped Trot

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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

    Authorized licensed use limited to: Khajeh Nasir Toosi University of Technology. Downloaded on December 21, 2009 at 05:11 from IEEE Xplore. Restrictions apply.

  • 7/25/2019 Control of a Quadruped Trot

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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