Foot Trajectory for a Quadruped Walking Machine

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  • 7/25/2019 Foot Trajectory for a Quadruped Walking Machine

    1/8

    IEEE

    International Workshop on Intelligent Robots and Systems

    IROS

    90

    F o o t

    T

    r

    aj

    c t

    o r y

    f o r

    a Q u a d r u p e d W a l k i n g Mach ine

    Y o s h i h i r o SA K A K IB A R A, K a z u t o s h i K AN , Y u u j i H OS ODA ,

    M a k o t o HA T TO RI a n d M a s a k a t s u F U J I E

    M e c h an i c a l E n g i n e e r i n g R e s ea r ch L a b o r a t o r y , H i t a c h i , L t d .

    5 0 2 K a n d a t u . T s u c h i u r a ,

    I

    b r a k i 3 0 0 , J a p a n

    A b s t r a c t

    T h i s p ap e r d e a l s w i t h t h e f o o t t r a j e c t o r y d e s i g n

    f o r a q i i a d r u p e d w a l k i n g m a c h i n e .

    A

    quadruped walk ing

    m a c h in e r e q u i r e s b o t h

    u n e v e n

    t e r r a i n w a l k i n g a n d h i g h

    - sp e ed f l a t s u r f a c e w a l ki ng c a p a b i l i t y .

    T h e

    s t a t i c

    walk in g method was used fo r u n e v e n t e r r a i n w a l k i n g

    and the dynamic walk ing method

    for

    f l a t p l a n e w a lk in g .

    I n

    t h e d y na m i c w a l k i ng c a s e , t h e r e l a t i v e s p ee d

    b e t w ee n t h e f o o t a nd t h e g r o u nd c a u s e s

    l o s s of'

    body

    b a l a n c e . f o o t t r a j e c t o r y i s d e s i g n e d b a se d on tw o

    p o i n t s , t h e k i n e m a t i c s o f f o o t m o ti on a nd t h e

    r e l a t i o n s h i p b e tw ee n j o i n t m ot io n a nd j o i n t d r i v i n g

    t o r q u e . T h i s pa p e r a l s o d i s c u s s e s a m e th od f o r ,

    r e d u c in g i m p a ct f o r c e up on i n i t i a l c o n t a c t w it h a

    f l o o r by a p e r i o d i c f o o t t r a j e c t o r y b a se d o n the wave

    motion of a cam. I n t h i s m e th o d, v e r t i c a l a n d

    h o r i z o n t a l m o t i o n of a f o o t t r a j e c t o r y w e re

    i n d e p e n d e n t l y g e n e r a t e d u s i n g c y c l o d i c m o t io n . T h i s

    t r a j e c t o r y w as d e s i g n a t e d t h e c om p o s i t e c y c l o i d f o o t

    t r a j e c t o r y .

    t

    1. I n t r o d u c t i o n :

    I n

    r e c e n t y e a r s t h e r e h a s b ee n a c t i v e d e ve l op m e n t

    i n r o b o t s w h ic h h av e l e g s t h a t a r e c a p a b l e

    of

    moving

    i n t h e s am e w o r k i n g e n v i r o n m e n t a s hu m an s. P l a c i n g

    a n i m p o r t an c e u po n p r a c t i c a b i l i t y

    i n

    p a r t i c u l a r , m a n y

    d e v e l o p m e n t s a r e b e i n g m ad e f o r q u a d ru p e d w a l k i n g

    mach ines Cll-C5l. T h e r e as o n f o r t h i s i s , a q u ad ru pe d

    m e ch an is m ha s g r e a t e r s t a b i l i t y w h i l e w o r ki n g

    c o m pa r ed w i t h a b i p e d a l m a c hi ne C 6 3 an d c o n s i s t s o f

    l e s s e l e m e n t s c o m p a r e d w i t h a h e x a p e d a l m e c h a ni s m .

    A

    h ig h l e v e l o f u n ev en t e r r a i n w a l k in g c a p a b i l i t y ,

    w hi ch i s o b t a i n e d fr om i t s l e g s , i s r e q u i r e d i ri a

    quadruped mechan i sm.

    I n

    o r d e r t o a t t a i n t h i s

    c a p a b i l i t y , f i r s t of a l l , s t u d i e s a r e b e in g do ne f o r

    s t a i r s w a lk in g C21 and fo r t u rn wa lk ing by means of

    s t a t i c w a lk in g C71 w h e r e i n m or e th a n t h r e e l e g s a r e

    i n c o n t a c t w i t h t h e g r ou n d a t o n e t i m e . On t h e

    o t h e r h a n d , the m a c h in e i s r ' e qu i re d t o w a lk a t h i g h

    s pe ed on a f l a t s u r f a c e . To a t t a i n t h i s h i gh s p e e d ,

    r e s e a r c h e r s a r e d e v e l o p i n g a d yn a m ic w a l k i n g m et h od

    w h er e o n l y t wo l e g s w i l l c o n t a c t t h e g r ou n d a t t h e

    same timeC81.

    F i g . 1. Sequent ia l pho tography o f

    compact qu adruped machine

    c o m p a ct q u a d r u p e d m a c h i n e h a s b e e n d e v e l o p e d a n d

    a s t u d y h a s b e e n d on e i n d y n am i c w a l k i n g u s i n g t h i s

    m a ch in e a n d v a r y i n g t h e g e o m e t r i c a l p a r a m e t e r s a nd

    w a l k i n g p e r i o d s of w a l k i n g p a t t e r n s .

    A s

    a r e s u l t , i t

    has bee ii p roven th a t

    the

    f o o t t r a j e c t o r y g r e a t l y

    i n f l u e n c e s s t a b i l i t y a nd e f f i c i e n c y o f w a lk i ng

    [SI.

    A prob lem i s c r e a t e d a s t h e w a l k i ng s p e e d or t h e

    m a c h i n e i n c r e a s e s .

    T h e

    i m pa ct f o r c e t h a t

    i s

    g e n e r a t e d by t h e f o o t c o n t a c t i n g w it h t h e f l o o r

    p r o d u c es b ad e f f e c t s u po n c o n t i n u o u s , s t a b l e w a l k in g

    a n d r e d u c t i o n o f l o a d a g a i n s t t h e m a c h i n e C I O I. As a

    s o l u t i o n t o t h i s pr ob le m , t h e c u r r e n t r e p o r t p r o p o se s

    a low i m p a ct w a l k i ng p a t t e r n w i t h p a r t i c u l a r

    a t e n t i o n o n f o o t m ot io n t r a j e c t o r i e s .

    2. W a l k i n g P a t t e r n G e n e r at i o n :

    T h e mot ion o f a walk ing robot i s d e t e r m i n e d by

    I n this

    r e p o r t , a t t e n t i o n is

    tw o f a c t o r s , t h e mo ti on t r a j e c t o r i e s a t t h e w a i s t C l l l

    andGa t t h e f o o t C121.

    p a i d t o t h e m o t io n o f t h e f o o t w h i c h h e l p s i n

    r e d u c i n g

    the

    r e a c t i o n

    of

    the f l o o r C127, a n d t h e

    m o ti on a t the w a i s t is a s s um e d t o b e u n i f o r m ;

    s t r a i g h t n o t i o n s t h a t p r e v e n t sh a k i n g o f m a t e r i a l

    b e in g c a r r i e d .

    O n e

    w a l k i n g p e r i o d

    i s

    d i v i d e d i n t o

    p h a s e s, , a s t a n c e p h a s e a nd a sw i n g p h a s e ,

    the

    prob lem

    -

    3 5 -

    Authorized licensed use limited to: Khajeh Nasir Toosi University of Technology. Downloaded on December 21, 2009 at 05:56 from IEEE Xplore. Restrictions apply.

  • 7/25/2019 Foot Trajectory for a Quadruped Walking Machine

    2/8

    l i e s

    i n

    hoH t o c o n r r o l t h e s t a r t a nd t h e s t o p

    o f the

    s * i n g m o t i on , s i n c e t h e s i a n c e p h a s e h as no mot ion

    a g a i n s t t h e f l o o r .

    A

    p r o b l e m e x i s t i n d e t e r m i n i r i g

    t h e t r a j e c t o r y w h ic h s a t i s f i e s t.h e r e q u ir e d s t e p

    l e n g t h a nd f o o t h e i g h t . E ve n i f t .he foot. height.

    r c a c he s i t s h i g h e s t p o i n t a t t h e i n t e r m e d i a te p o i nt ,

    o f t h e s w i n g , t h e t r a j e c t o r y p r o b l e o f c o n n e c t i n g t h e

    t h r e e p o i n t s , i n c l u d i n g t h e i n te r m e d i a t e p o i n t s t i l l

    e x i s l . T h e r e h a ve b e en a few methods proposed

    r e g a r d i n g t r a j e c t o r y f o r m a n i p u l at o rs

    C131.

    Y e t ,

    i n

    c o n v e n t i o n a l f o o t t r a j e c t o r y f o r a w a l k i n g r o b o t , t h e

    w a l k i n g sp e e d was s l o w , c r e a t e d l e s s i m p a ct a g a i n s t

    t h e f l o o r , a nd c r e a t e d n o f o o t m o ti on c o n f i g u r a t i o n

    o r s p e e d p a t t e r n p r o b le m . H o we v er , t h e qu a d r up e d

    w a l k i n g m ac h i ne , w h ic h i s t h e s u b j e c t of t h e p r e s e n t

    s t u d y , h a s a m as s o f a p p r o x i m a t e l y 300 k g an d i s m ade

    t o m ove a t a d y n am ic w a l k i n g s p eed o f 2 . 5 k m / h o u r .

    When t h e s i m u l a t i o n ex p e r i m en t s were co n d u c t e d

    a p p l y i n g t h e w a l k i ng p a t t e r n s d e v e l op e d f o r t h e

    c o m pa c t qu a d r up e d r o b o t , i t w as o b s e rv e d t h a t t h e

    r o b o t l o s t i t s b a l a n c e when t h e f o o t i m p ac t e d w i t h

    t h e g ro u nd a n d s t a b l e w a l k in g p a t t e r n s w e re g r e a t l y

    d i s t u r b e d .

    d e v e lo p m e n t o f w a l k i ng p a t t e r n s t a k i n g i n t o

    c o n s i d e r a t i o n t h e a n a l y s i s o f f o o t m ovem ent a n d t h e

    d y na m ic a n a l y s i s o f t h e l e g s .

    T he g a i t a n d w a lk i n g p a t t e r n s i n c o n s t a n t s p ee d

    s t r a i g h t w a l ki ng a r e d i s c u s s e d b e l ow .

    I n

    t h e c a s e o f

    a q u ad ru p ed ro b o t , t wo eg s m ak e a p a i r and each p a i r

    h a s

    a g a i t of a l t e r n a t e l y r e p e a t i n g t h e s t a n c e a nd

    s w i n g . When t h i s g a i t i s a d o p t e d , t h e m o ti o n p a t t e r n

    o f a s i n g l e l e g j o i n t c a n be a p p l i e d t o o t h e r l e g s ,

    an d o n e cy c l e o f t h e q u ad ru p ed m ov em en t can b e

    c o n s t r u c t e d i n t w o p h a s e s .

    I n

    t h e p r e s e n t s t u d y , t h e

    t r o t t i n g g a i t o u t o f s e v e r a l g a i t s f r om a p a i r o f

    l e g s i s a d a p t e d .

    T h e r e f o r e , t h e p r e s e n t s t u d y t a c k l e d t h e

    F i 9.2. Q uadruped wa

    I k

    i ng mach

    i

    n e

    2 . 1 W a l k i n g P a t t e r n G e n e r a t i o n M e t h o d :

    Two d i f f e r cn t v iewp o i n 1 .s

    a r e I ISP( I 10

    program

    i n o r f

    - l i n e t h e w a l k in g p a t t e r n of a w a l k i n g r o b o t .

    1 ) A

    m e th o d t h a t c o n s i d e r s f o o t m o t i o n :

    A

    mrr.hod to

    d e t e r m i n e

    3

    p o i n t s , i . e . t h e d e p a r t i n g p oi n t. w he re

    a f o o t l e a v e s t h e f l o o r , t h e m aximum f o o t h c i g l i t

    p o i n t w h e re t h e f o o t r e a c h e s t h e h i g h e s t p o s i r i o ri

    a rid t h e l a n d i n g p o i n t , a nd a t r a j e c t o r y c o n n e c t i n g

    t h e s e t h r e e p o i n t s .

    2 )

    A

    m e th od t h a t c o n s i d e r s t h e m o t io n a nd f o r c e o f

    a

    j o i n t :

    A

    m et ho d t o d e t e r m i n e j o i n t m o t i o n s p a y in g

    a t t e n t i o n t o t h e r o t a t i n g s p ee d an d t o rq u e of a

    j o i n t .

    T he w a lk in g v e l o c i t y ( s t e p 1engt.h a n d w a l k i n g p e r i o d ) ,

    f o o t h e i g h t , a nd w a i s t h e i g h t w e re u se d t o d e t e r m i n e

    t h e p ar a m e t e r s f o r a w a l k in g p a t t e r n o n a f l a t s u r f a c e .

    i n

    d e t e r m i n i n g a c c u r a t e f o o t h e i g h t a nd j u d g i n g

    i n t e r f e r e n c e w it h t h e o u t s i d e e nv i ro n m e n t d u r i n g a

    swing, method

    1

    i s co n s i d e r e d s u i t a b l e . I lo w ev e r,

    i n

    o r d e r t o s h o r t e n t h e w a lk i n g p e r io d t o i m p ro ve t h e

    wa l k i n g s p eed , m e t h o d

    2 )

    i s b e t t e r s u i t e d , s i ri ce i t

    i s do n e i n r e l a t i o n s h i p w i t h t h e m aximum a n g u l a r

    v e l o c i t y , maxim um a n g u l a r a c c e l e r a t i o n , a nd

    m a x i m um

    t o r q u e

    of

    a s w in g in g j o i n t . D u ri ng f l a t s u r f a c e

    w a l k i n g , a c c e l e r a t i o n a nd d e c e l e r a t i o n o f t h e

    swin ging le g mot ions can he programmed ahead of t ime

    t a k i n g i n t o c o n s i d e r a t i o n t h e am o un t o f c o n t a c t .

    b e tw e e n t h e f l o o r a n d t h e f o o t i n m o t i o n.

    i n

    t h e

    p r e s e n t s t u d y , t h e a u t h o r s f i r s t d ev e lo p ed t h e l e g

    w a lk in g p a t t e r n s d e t e r m i n e d

    b y

    fo o t m o v em en t s b as ed

    on t h e g r ou n d c o o r d i n a t e s . T h e se c a n e a s i l y d e s c r i b e

    t h e m ot io n s n e a r t h e c o n t a c t p o i n t s w i th t h e f l o o r ,

    a nd a l s o ba s ed o n t h e c o o r d i n a t e s f i x e d

    on

    the body

    o f t h e r o b o t ( f i x e d w a i s t c o o r d i n a t e ) .

    As

    t h e s p eed

    a c c e l e r a t e s , a s m oo th a c t i o n i s r e q u i r e d o f t.h e

    wa l k i n g ro b o t . T h en an a l y s i s was m ad e t o t h e wa l k in g

    p a t t e r n d e v e l o p e d i n t h e a b o v e m an n er

    b y

    method

    Z ) ,

    a s i t r e q u i r e s i n s t r u c t i o n s c o n s i de r i ng t h e

    g e n e r a t i o n of i n e r t i a f o r c e by . j o i n t m o v em e n ts . t h e

    r e s t r i c t ed d y n am i cs o f j o i n t m ov emen ts an d b od y

    s h a k i n g .

    2 . 2 W al k i ng P a t t e rn Dev e l o p m en t Dev i ce :

    T he p r o c e s s f l o w o f t h e c u r r e n t w a l k in g p a t t e r n

    d ev e l o p m en t i s s h o wn

    i n

    F i g .

    3 .

    F i r s t , t h e

    s p e c i f i c a t i o n d a t a o f t h e sp e ci m en r o b o t a r e e n t e r e d .

    I n

    c o n c r e t e t e r m s , t h e y i n c l u d e t h e d im e n s i o n s of

    l i n k s , m a s s

    of

    l i n k s a n d t h e p o s i t i o n o f t h e ma ss

    ( c e n t e r of g r a v i t y o f l i n k s ) i n a m a t e r i a l p a r t i c l e

    m o d el . N e x t , p a r a m e t e r s , s u c h

    a s . foot

    h e i g h t . s t e p

    l e n g t h , w a i s t h e i g h t , a n d w a l k i n g p e r i o d s n e c e s s a r y

    t o

    d e t e r m i n e t h e f o o t t r a j e c t o r y a n d w a i s t p o s i t i o n

    t r a j e c t o r y a r e i n p u t .

    Fn

    a n a l y z i n g t h e f o o t m o t i o n s . t h e m ov em en ts a r e

    a n a ly z e d by g r a p h i c d i s p l a y o f v a r i a t i o n s

    of

    t h e f o o t

    - 316

    -

  • 7/25/2019 Foot Trajectory for a Quadruped Walking Machine

    3/8

    Robot Specifications

    *

    Walking Paltern Parameter Input

    Foot Height Max.

    Waist Height

    Period Normal)

    Data Input

    Step

    Foot

    Trajectory

    Analysis of Foot Motion

    Relative Motion

    Velocity. Acceleration

    to the Floor

    Waist Trajectory

    Analysis of Body Motio n

    &

    I

    Memo N lor Joi nt Anele Reference

    I

    Joint Angle Calculator

    Continuity

    Maximum Angle Velocity

    Maximum Angle Acceleration

    Analysis of Join t Motion

    Reference

    Dynamics Analysis

    Joi nt Torque

    F i g . 3 .

    Process

    o f

    w a l k i n g

    p a t t e r n g e n e r a t i o n

    p o s i t i o n , v e l o c i t y and a c c e l e r a t i o n . T he f o o t

    t r a j e c t o r y dat .a and w a i s t p o s i t i o n t r a j e c t o r y , t h us

    d et er mi ne d, a r e p u t i n t o t he j o i n t a n gl e c a l ~ u l a t i n g

    s e c t i o n a nd a r e c o nv e r t ed i n t o t a r g e t j o i n t a ng l es .

    T he se a r e a n a l yz e d i n t h e j o i n t a n g u l a r m o ti o n

    a n a l y s i s s e c t i o n . On e ac h j o i n t , t h e j o i n t a n g l e s ,

    a n g ul a r v e l o c i t y , a nd a n g ul a r a c c e l e r a t i o n s a s v a r i e d

    b y t ime p assag e a nd a r e g ra p h i ca l l y d i sp l a ye d an d th e

    co n t i n u i t y a n d ma xi fl um va lu e s a re ch e c te d .

    v a r i a t i o n s a r e o b t a i n e d b y d y n am i c a n a ly s e s of these

    j o i n t a n g u la r mo t i o n s . Th ese a n a l yse s we re r e p ea te d ,

    th e r e s u l t s were s tu d ie d , a nd th e wa l k i n g p a t te r n was

    g e n er a t ed . T he f i n a l d a t a g e n e ra t e d f o r t h e t a r g e t

    j o i n t a n g l e s a r e s t o r e d i n t h e m emory.

    I t

    a r e t h e n

    o u t p u t t o a j o i n t a c t u a t o r on t h e r o b o t b y a w a l k i ng

    c o n t r o l d e v i c e ac c o r di n g t o t h e a c t u a l w a l k i n g p e r i o d ,

    and the ro bot wa lks.

    Torque

    3 .

    Wa lk i n g P a t te rn s :

    Leg mot ions of a quadruped wal k i ng ro bot can be

    d e s c r i b e d b y t he p o s i t i o n s

    of

    t h e f o u r f e e t a nd t h e

    p o s i t i o n o r o r i e n t a t i o n of t h e b o d y ce n te r a s

    d e t er m i ne d d u r i n g t h e p as sa ge o f t i m e. I n t h i s p a pe r

    a t t e n t i o n i s p a i d t o t h e m t i o n s o f e ac h le g . The

    moment when th e fo o t l e a ve s th e f l o o r i s e xp re sse d a s

    t = O ,

    the l and ing moment as t ime TY, and the nex t

    moment o f dep ar t ure as T. I n o t h e r wo rds , o n e

    w a l k i n g p e r i o d w i l l be T, the swin g ing pe r i od Ty, and

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

    T -

    Ty=Tr. When the ti me t

    i s

    T y

    -0.25T O T 0 . 2 5 T 0 . 5 T 0

    5 T

    3

    - 4 1 1

    - 0 . 2 5 T O T 0 . 2 5 T 0 . 5 T 0 . 7 5 T

    I

    t i m e ( T = 0 . 7 s e c )

    6 ) Z axis

    accelerat ion

    Fi9.7. Sinusoidal

    f o o t

    motion

    -

    3 8

    -

  • 7/25/2019 Foot Trajectory for a Quadruped Walking Machine

    5/8

    t he x axi s. As t he boi i ndary condi t i on, the sw ng l eg

    moi . i on accel erat i on i s assumed a s 0 at. t he connect i ng

    pui nl

    w 1 . h

    t he sI.;ince

    pe r i o d . Then,

    .he

    acclclcvat

    i on

    i n

    one swi ngi ng peri od i s vari ed si nusoi dal l y and

    express ed

    a s

    X

    = A m

    s i n n t / Ty

    ( 3 )

    A speed of 0 i s assumed at t he connecti ng poi nt

    bet ween t he sw ngi ng leg peri ods and t he st ance

    peri ods as the boundary condi t i on for the vel oci ty.

    When X =

    0

    at t =

    0,

    and X =

    0

    at. t = T y,

    . Am T y

    t

    1

    ( 4 )

    . ___ I-cos 2 n __

    and the posi t i on i s obt ai ned by i ntegrai i ng

    2 n TY

    Equat i on ( I ) i t h t

    Z = Am2 si n n __

    TY

    The.speed i s obtai ned by i ntegrat i ng Eqi i ati on ( 1 0 )

    when Z=O a t . 1.=0 Ty J 2

    b y

    *

    Am2 Ty t

    2 = I _

    ( 1

    -

    cos 4n-) 1 1 )

    4 K TY

    The posi t i on i s expressed by i ntegrati ng

    Equaf i on ( 1 1 ) w t h t ,

    when z=O at t =O and z=llo at t =Ty/ 2 by

    TY

    m2 Ty Ty t

    s i n 4 -

    )

    (12)

    - -

    __

    =

    4 n 2

    4 n

    TY

    8 Ilo

    TY

    Am2 =

    Am Ty

    TY

    t s i n

    x t

    c1

    Then Equat i on ( 1 3 ) i s assi gned to Equati ons 101,

    (111 ,

    z -

    and (12) to obt a i n the f o l l owi ng,

    2 x n

    TY

    When X=O at t =O, and X=So at t =Ty

    n so

    TY

    C1

    =

    0, Am1 =

    _ __

    Equat i on

    (6)

    i s assi gned to Equati ons ( 3 1 , ( I ) , nd

    ( 5 )

    to obt a i n the f o l l owi ng,

    2 nSo

    t

    x = - s i n 2 n ( 7 )

    TY

    TY

    s o

    t

    TY TY

    x = -1

    -

    cos z n -

    1 t

    T Y 27t TY

    X

    =

    So(-

    -

    __

    s i n 2n - )

    1

    =

    So( tn -

    __

    s i n 2 n t n )

    ( 9 )

    n

    I n z axi s movements, t he boundary condi t i ons of t he

    f oot moti ons are set s that the accel erat i on and

    vel oc i ty wi l l bec ome

    0

    at poi nts of depart ure,

    maxi mum f oot hei ght and l andi ng. Regar di ng the

    peri odi c moti ons of a f o o t r epedt i ng i mpi ngement upon

    the fl oor as a t ype of ver t i c al mot i ons of

    a

    shaf t by

    a cam the accel erat i on i s var i ed s i nusoidal l y .

    Agai nst

    0

    5 E S T y / 2

    .

    2110

    t

    z

    =

    1 -

    c o s

    4 n

    ___ )

    TY TY

    ( 15)

    t 1 t

    2 =

    2110

    (-

    -

    __

    s i n 4n-

    )

    ( 1 6 )

    TY 4 n

    TY

    1

    =

    ZHo( t n

    -

    ___

    s i n

    n

    tn )

    4 n

    N

    100

    200 300 400

    X

    (mm)

    (1) G r o u n d c o o r d i n a te f o o t t r a j e c to r y

    I 1

    N

    O Z

    00 200 300 400 X

    (mm)

    0

    (2)

    W ai s t f i x e d c o o r d i n a te f o o l t r a j e c to r y

    F i 9 . 8 .

    Composite cycloid foot

    trajectory

    b

    - 3 9 -

    Authorized licensed use limited to: Khajeh Nasir Toosi University of Technology. Downloaded on December 21, 2009 at 05:56 from IEEE Xplore. Restrictions apply.

  • 7/25/2019 Foot Trajectory for a Quadruped Walking Machine

    6/8

    .- - 5 0 0 1 I ,

    .

    -1000

    -0.257-

    OT 0.25T

    0.5T

    0.75T

    m

    E 800-

    0 -

    .-

    -800-

    .5

    -1600

    -

    time (T=0.7sec)

    ( I )

    X

    axis posit ion

    I I I

    time (T=0.7sec)

    (2) x

    axis velocity

    , -

    J N

    i 41 I

    F i g . 9 .

    C o m p o s i t e c y c l o i d ' f o o t m o t i o n

    Shown iii F i g . 8 a r e t h e f o o t t r a . j er t . o ri e s e x i i rp c -

    by the ahove formulas

    a s

    a p p l i c d 1 .0 nii a c t . i i a 1 rob.-

    l,'ip,iir I i i i r l i i : . ~ ~ c s

    11v

    I I I C I I

    i r j l i s i n x

    :11 i r 1 7 (1 i r r r . l

    i '

    va r i e d h y t ime . S w in g in g 1r:g p er io il c ~ ~ r v c sn 1 . 1 ~ :

    1 ) a n d ) ( I ) i l l g i v e c y c l o i i l C u r v e s , and thc l ' o r .

    l r a j ec to ry cxprc:ssc:d

    t i y

    groi11111 oortliniiLes

    w i l l

    lic

    composi t , ion o f r o o t mot . io ns i n ( l i e

    x

    arid 7 d i r e c t i c -

    l t i e re fo r c , wc

    s h a l l

    c a l l I . h i s

    l o o t

    t . ra , iectory a

    composi l e cy c l o i d t , ra je ctor y . An improvement.

    call .

    noted where , i n compar ison

    L O t.he

    s i n ~ i s o i d a lwave

    t r a j e c t o r y i n t ,he pr e v i oi i s s e c t i o n , t h e v e l o c i t y

    a r m -

    a c c e l e r a t i on h e r e c an v a r y c o n t i n u o r ~ s l ywhen t.-0 a r

    =Ty=0.5T. I f t.he f o o t v e l o c i t y i s co n t i n u o i i s a t i i , . -

    s h i f t i n g p e r i o d b e lwe en th e s t ,a nce an d sw in g in g

    mo l i o n s , th e mot . io n o f t h e legs w i l l be smooth. A i

    s i n c e the b o ~ in i l a r yarea t)et.ween

    L h c

    st .anr l ing foo l

    mot ion on Lhe f l o o r cl i i r i r ig

    a

    s ta n ce p e r i o d a n d the

    s w i n g i n g

    mor. ion

    i s smo0t.h a n d c o r i ~ i n i i o ~ r s ,

    he Too .

    c an b e p l a c e d on t h e f l o o r o r l i f t e d f ro m th e f l o o r

    a t t h e r e 1a t .i v e v e l o c i t y of z p r o ,

    t h u s

    rcd i ic in l :

    imp a c t a t the t,ime o f

    c o l i t a c t .

    time (T=0.7sec)

    ( I )

    acceleration (sinusoid)

    t ime (T,=0.7sec)

    (2)

    acceleration (composite cycloid)

    F i g . 1 0 .

    F o o t m o t i o n f o r v a r i o u s

    t r a e c t o r i es

    We h a ve t .h us fa r d e sc r i b e d two typ e s o f f o o t

    t r a j e c t o r i e s . T he a ii t .h or s a r i a l y z c d t h e p u s i t i o r i ,

    v e l o c i t y , a nd ac c e l e r a t i o n o f e a c h f o o t t ra.iPc:tory.

    d e ve lo p e d a new t r a j e c t o r y , o b ~ a i n e d s mo ot he r

    ve lo c i ty and acce lera t . ion , and reduced thc maximum

    ve lo c i t y an d ma ximu m a cc c r l e ra t i o n . The re su l . a n t

    a c c e l e r a t i o n s

    of

    t h e foot i n the vert , ic:a l and

    h o r i z o n t a l d i r e c t i o n s i n d i c a t e d a g a i n s t t h e t i me a x is

    a r e s ho wn i n F i g . l O .

    I n t h e c o m p os i te c y c l o i d i n F i g . 1 0 ( 2 ) . f o o t

    a c c e l e r a t i o n s s ho w c o n t i n u o u s v a r i a t i o n s f r o m t h e

    s t a n ce p o s i t i o n .

    - 320 -

  • 7/25/2019 Foot Trajectory for a Quadruped Walking Machine

    7/8

    4 .

    A n a l y t i c a l R e s u l t s o f J o i n t A n g u l a r M o t i o n s

    o f

    L e g W a l k i n g P a t t e r n s :

    20

    .- %

    5 4

    2 % 0

    ,"E -10-

    10-

    0

    E -20

    T he p r e v i o u s c h a p t e r d i s c ~ r s s e d t h e d e v e l o p m e n t of

    w a l k i ng p a t t e r n s f ro m t h e v i e w p o i n t

    of

    f o o t m o t i o n s

    a n d t h e i r e v a l u a t i o n s .

    per fo rmed b y a n a c t u a t o r a t t a c h e d t o e ac h j o i n t .

    T h e r e f o r e , i n t h i s c h a p t e r , t h e w a l ki n g p a t t e r n s

    d e s c r i b e d a r e c o m p a re d an d s t u d i e d f ro m t h e v i e w p o i n t

    of j o i n t m o ti o n s. T he t o r q ue a p p l i e d t o a j o i n t

    v a r i e s d e p e n d in g on t h e w e ig h t d i s t r i b u t i o n on each

    l i n k . T h e r e f o r e , t h e l e g m a ch in e p r e p a r e d f o r t h i s

    s t u d y was f i r s t m ade i n t o a s i m u l a t i o n m o de l o f a

    c o n c e n t r a t e d p a r t i c l e s y s te m . F i g .

    5

    shows the

    s i m u l a t i o n m o d e l . T h e p a r t i c l e m o de l wh ic h

    s i m p l i f i e s t h e s t a n c e mo de l a nd s w i n g mo de l a r e sh ow n

    aL Fig. 5. A t o r q u e c a l c u l a t i o n w a s m a d e o n t h i s

    model .

    F i g u r e 1 1 s ho ws t h e a n a l y t i c a l r k s u l t s of

    w a i s t a n d knee j o i n t m o t i o n s fo r t h e c o m p o s i t e

    The l eg movemen t s a re

    -10

    ;

    :i i

    9

    -40

    -

    -0.25T OT 0 .25T 0 .5T 10 .75T

    .g -80

    tim e ( T ~ 0 . 7 s e c )

    ( I ) waist joint angle

    -8001

    .s

    - -0.25T OT 0.25T 0.5T 0.75T

    t i m e ( T =0 . 7 s e c )

    (2)

    waist joint angle velocity

    c y c l o i d t r a j e c t o r y .

    W i t h

    regards t o a c r e l e r a t i o n ,

    t h e fo rm of a c c e l e r a t i o n v a r i e d

    i n

    t h e v e r t i c a l

    d i r e c t i o n

    irl

    a c o m p o si t e c yc l o i d t r a j e c t o r y

    ( F i g .

    9 ( 6 ) )

    v a r i a t i o n

    of

    t h e kn e e j o i n t ( F i g .

    t l ( 7 ) ) .

    At

    the

    s am e t i m e , t h e w a i s t a n g u l a r v e l o c i t y v a r i a t i o n ( i ' i g .

    l l ( 2 ) ) a l s o sh ow s i n f l e c t i o n p o i n t s . One . ca n

    rec og n iz e an improvemen t i n t h e f o l l o w - u p p e r f o r m a n c e

    of

    j o i n t m o ti o ns d u r i n g s w i n g in g , r e d u c t i o n of

    g e n e w t i o n of i n e r t i a b y s w i n g i n g t h e l e g , a n d

    r e d u c t i o n o f b od y s h a k i n g .

    a p p e a r s e x a c t l y t h e s am e a s ' t h e a c c e l e r a t i o n

    5.

    C o n c l u s i o n :

    I n

    d e v e l o p i n g t h e w a l k i n g p a t t e r n s f o r a w a l k in g

    r o b o t , i n f l u e n c e on t h e s t a b i l i t y of a r o b o t ' s

    w a l k i n g

    by

    f l o o r r e a c t i o n s w he n a f o o t c o me s

    i n

    c o n t a c t w it h t h e f l o o r w er e t a k i n g i n t o a c c o u n t .

    S t u d i e s w e r e d o n e f o r l e s s i m p a c t i n g f o o t m o t i o n s .

    C o n v e n t i o n a l l y , w a l k i ng p a t t e r n s h a v e b e en d e v e l op e d

    .s -80

    tim e ( T ~ 0 . 7 s e c )

    5 ) knee joint angle

    > -

    U

    5 -400

    -800

    t i m e ( T =0 . 7 s e c )

    6 )

    knee joi nt angle velocity

    t

    t i m e ( T =0 . 7 s e c )

    .-

    0

    (3)

    waist joint angle acceleration

    _

    t i m e ( T ~ 0 . 7 s e c )

    -

    0

    (7)

    knee joint angle acceleration

    _

    0

    -

    -80

    80

    - -0.25T OT 0.2 5T 0.5T 0.75T

    6

    -160

    t i m e ( T =0 . 7 s e c )ime

    *

    ( T = 0 . 7 s e c )

    (8)

    knee joint torque

    4) waist joint torque

    F i g . 11.

    J o i n t mot i o n

    - 32 -

    Authorized licensed use limited to: Khajeh Nasir Toosi University of Technology. Downloaded on December 21, 2009 at 05:56 from IEEE Xplore. Restrictions apply.

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    8/8

    from

    studi es

    of

    t he

    rorms

    and t he met hod

    of

    thei r

    i nterpol at i on.

    W t h

    thi s i n m nd, i l i s i mport an^ to

    rcmemher r h a t ,

    as

    t h e

    wal ki ng speed i ncreases, a

    conti nuit y exi sts between t he rel ati ve vel oci ty at

    t he f l oor and foot and the vel oci t y/ accel erati on at

    the f oot and j oi ni angl es. As a resul t , a composi t e

    cycl oi d tr aj ectory i s devel oped by expandi ng t he

    cycl oi d moti ons i nto two di mensi onal moti ons w t hi n

    I

    and z planes. W 1 . h thi s t r a j ecto ry , t he

    foot,

    can be

    set down o r l i f t ed f rom the f l oor at a re l at i ve speed

    of 0, and accel erat i on and decel er at i on can he

    conducted smoot hl y dur i ng sw ngs. Thus, the

    poss ib i l i t y of a f oot moti on w t h a smal l er maxi mum

    accel erati on has been cl ari f i ed. From t he vi ewpoi nt

    of j oint angular moti ons, the accel erati on of a

    composi t e cycl oi d t raj ectory

    i n

    a vert i cal di rect i on

    has been discussed. Thi s tr aj ectory has cl ari f i ed

    the poss i bi l i ty of maki ng t he act uat or move smoothl y

    w t hout burden and r educi ng the i mpact agai nst t he

    f l oor .

    Thi s i s cont r acted research under a maj or pr oj ect

    of

    t he Agency

    of

    I ndustr i al Sci ence and Technol ogy,

    MI TI , ent i t l ed the Advanced Robot .

    R efe r enc es

    C

    H. Ki mura, l . Shi moyama and H. M ura, Condi t i ons

    of

    Gai t Sel ecti on i n Dynam c Wal k

    of

    t he Quadruped,

    Proc.

    of

    5th Annual Conf . Roboti cs Soci ety

    of

    J apan,

    pp369-370, 1987.

    C21 S. Hi r ose,

    A

    Study of Desi gn and Cont r ol of a

    Quadruped Wal ki ng Vehi cl e, The I nternati onal

    J ournal

    of

    Roboti cs Resear ch, Vo1. 3, No. Z Summer

    1984.

    Quadruped Wal ki ng Machi ne, I CAR Pr oc. , pp65- 76 Sept.

    1987.

    C31 M Fuj i e,Y. Aosoda et al . , Devel opment of

    C41 McGhee, R. B and A. A. Fr ank, On t he Stabi l i t y

    Propert i es of Quadruped Cr eepi ng Gai t s,

    Mathemati cal Bi osci ences. 1968.

    E51 M I l . Rai bert et al . , Runni ng on Four Legs As

    Though They Wer e One, I EEE J our nal of Roboti cs

    and Autonation,Vol.RA.Z,No.2 1986.

    Walki ng Robot s, The f l i tachi l l youron, Vol . 68, No. l O,

    C61

    l . Katoh, M Fuj i e et al . , Devel opment of Legged

    ppZ5- 30, 1986.

    C71 M omr Vukobrat ovi c , Legged Locomoti on Robot s,

    C81

    M Fuj i e, Y. Sakaki bara et al . , Devel opment

    of

    Quadr upedal Mechani sm 1) St udy of Redundant

    Freedom Hechani srn Contr ol , Proc. of 6t h Annual

    Conf . Roboti cs Soci ety of J apan, pp321-324, 1988.

    [91

    K. Kan and M Fuj i e, Devel opment of M ni atur e

    Quadruped Robot - - A Study on Gai t w t h Wal k

    Exper i ment- - , P roc.

    of

    5th Annual Conf . Roboti cs

    Soci ety of J apan, pp365-366, 1987.

    1975.

    Cl01 Y. Hosoda, M Hattori et al . , Devel opment of

    Quadr uped Wal ki ng Machi ne( 3) Desi gn

    of

    Wal ki nn

    Mechani sm Proc.

    of

    6th Annual Conf . Roboti cs

    Soci ety of J apan, pp307-308, 1988.

    C l 1 1 Y. Sakaki bara, M Hattori et al . , Development o f

    Quadr uped Wal ki ng Machi ne( 6) Gr avi l y Center

    Traj ector y Generat i on f or Quadruped wal ki ng

    Machi ne, Proc. of

    7th

    Annual Conf . Roboti cs

    Soci ety of J apan, pp701-702, 1989,

    Quadr uped Dynam c Wal ki ng Machi ne( 2) Eval uati on

    of the Fl oor Reacti ng Force and Foot Tr aj ectory

    w th Si ngl e Leg Exper i ment , Proc. of 6th

    Ann i i a ;

    Conf . Roboti cs Soci ety of J apan, pp325-328,1988.

    Cl 31

    I l . Ozaki , M Yamamoto and A. Mohr i , Pl anni ng

    of

    Near- M ni mumTi me J oi nt Traj ectory f or

    Mani pul ators Usi ng B- Spl i ne, Trans. of The

    Soci ety of I nstr ument and Cont rol Engi neers, Vol . -

    N0. 11, pp83- 89, 1987.

    C121 Y. Sakaki bar a, K. Kan et al . , Devel opment o f

    - 3 -