Foot Trajectory for a Quadruped Walking Machine
Transcript of 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.
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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
-
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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
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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
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.- - 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 .
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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
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7/25/2019 Foot Trajectory for a Quadruped Walking Machine
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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
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